text: :: TOPREALC semantic presentation begin theorem :: TOPREALC:1 for X being trivial set for Y being set st X,Y are_equipotent holds Y is trivial proof let X be trivial set ; ::_thesis: for Y being set st X,Y are_equipotent holds Y is trivial let Y be set ; ::_thesis: ( X,Y are_equipotent implies Y is trivial ) assume A1: X,Y are_equipotent ; ::_thesis: Y is trivial A2: card X < 2 by NAT_D:60; A3: Y is finite by A1, CARD_1:38; card X = card Y by A1, CARD_1:5; hence Y is trivial by A2, A3, NAT_D:60; ::_thesis: verum end; registration let r be real number ; clusterr ^2 -> non negative ; coherence not r ^2 is negative ; end; registration let r be positive real number ; clusterr ^2 -> positive ; coherence r ^2 is positive ; end; registration cluster sqrt 0 -> zero ; coherence sqrt 0 is empty by SQUARE_1:17; end; registration let f be empty set ; clusterf ^2 -> empty ; coherence sqr f is empty ; cluster|.f.| -> zero ; coherence |.f.| is empty by RVSUM_1:72; end; theorem Th2: :: TOPREALC:2 for c1, c2 being complex number for f being complex-valued Function holds f (#) (c1 + c2) = (f (#) c1) + (f (#) c2) proof let c1, c2 be complex number ; ::_thesis: for f being complex-valued Function holds f (#) (c1 + c2) = (f (#) c1) + (f (#) c2) let f be complex-valued Function; ::_thesis: f (#) (c1 + c2) = (f (#) c1) + (f (#) c2) A1: dom (f (#) (c1 + c2)) = dom f by VALUED_1:def_5; A2: dom (f (#) c1) = dom f by VALUED_1:def_5; A3: dom (f (#) c2) = dom f by VALUED_1:def_5; A4: dom ((f (#) c1) + (f (#) c2)) = (dom (f (#) c1)) /\ (dom (f (#) c2)) by VALUED_1:def_1; hence dom (f (#) (c1 + c2)) = dom ((f (#) c1) + (f (#) c2)) by A2, A3, VALUED_1:def_5; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom (f (#) (c1 + c2)) or (f (#) (c1 + c2)) . b1 = ((f (#) c1) + (f (#) c2)) . b1 ) let x be set ; ::_thesis: ( not x in dom (f (#) (c1 + c2)) or (f (#) (c1 + c2)) . x = ((f (#) c1) + (f (#) c2)) . x ) assume A5: x in dom (f (#) (c1 + c2)) ; ::_thesis: (f (#) (c1 + c2)) . x = ((f (#) c1) + (f (#) c2)) . x hence (f (#) (c1 + c2)) . x = (f . x) * (c1 + c2) by VALUED_1:def_5 .= ((f . x) * c1) + ((f . x) * c2) .= ((f (#) c1) . x) + ((f . x) * c2) by A1, A2, A5, VALUED_1:def_5 .= ((f (#) c1) . x) + ((f (#) c2) . x) by A1, A3, A5, VALUED_1:def_5 .= ((f (#) c1) + (f (#) c2)) . x by A1, A2, A3, A4, A5, VALUED_1:def_1 ; ::_thesis: verum end; theorem :: TOPREALC:3 for c1, c2 being complex number for f being complex-valued Function holds f (#) (c1 - c2) = (f (#) c1) - (f (#) c2) proof let c1, c2 be complex number ; ::_thesis: for f being complex-valued Function holds f (#) (c1 - c2) = (f (#) c1) - (f (#) c2) let f be complex-valued Function; ::_thesis: f (#) (c1 - c2) = (f (#) c1) - (f (#) c2) thus f (#) (c1 - c2) = (f (#) c1) + (f (#) (- c2)) by Th2 .= (f (#) c1) - (f (#) c2) by VALUED_2:24 ; ::_thesis: verum end; theorem Th4: :: TOPREALC:4 for c being complex number for f, g being complex-valued Function holds (f (/) c) + (g (/) c) = (f + g) (/) c proof let c be complex number ; ::_thesis: for f, g being complex-valued Function holds (f (/) c) + (g (/) c) = (f + g) (/) c let f, g be complex-valued Function; ::_thesis: (f (/) c) + (g (/) c) = (f + g) (/) c A1: dom ((f (/) c) + (g (/) c)) = (dom (f (/) c)) /\ (dom (g (/) c)) by VALUED_1:def_1; A2: dom (f (/) c) = dom f by VALUED_2:28; A3: dom (g (/) c) = dom g by VALUED_2:28; A4: dom ((f + g) (/) c) = dom (f + g) by VALUED_2:28; A5: dom (f + g) = (dom f) /\ (dom g) by VALUED_1:def_1; thus dom ((f (/) c) + (g (/) c)) = dom ((f + g) (/) c) by A1, A2, A3, A4, VALUED_1:def_1; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f (/) c) + (g (/) c)) or ((f (/) c) + (g (/) c)) . b1 = ((f + g) (/) c) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f (/) c) + (g (/) c)) or ((f (/) c) + (g (/) c)) . x = ((f + g) (/) c) . x ) assume A6: x in dom ((f (/) c) + (g (/) c)) ; ::_thesis: ((f (/) c) + (g (/) c)) . x = ((f + g) (/) c) . x thus ((f + g) (/) c) . x = ((f + g) . x) / c by VALUED_2:29 .= ((f . x) + (g . x)) / c by A1, A2, A3, A6, A5, VALUED_1:def_1 .= ((f . x) / c) + ((g . x) / c) .= ((f . x) / c) + ((g (/) c) . x) by VALUED_2:29 .= ((f (/) c) . x) + ((g (/) c) . x) by VALUED_2:29 .= ((f (/) c) + (g (/) c)) . x by A6, VALUED_1:def_1 ; ::_thesis: verum end; theorem :: TOPREALC:5 for c being complex number for f, g being complex-valued Function holds (f (/) c) - (g (/) c) = (f - g) (/) c proof let c be complex number ; ::_thesis: for f, g being complex-valued Function holds (f (/) c) - (g (/) c) = (f - g) (/) c let f, g be complex-valued Function; ::_thesis: (f (/) c) - (g (/) c) = (f - g) (/) c thus (f (/) c) - (g (/) c) = (f (/) c) + ((- g) (/) c) by VALUED_2:30 .= (f - g) (/) c by Th4 ; ::_thesis: verum end; theorem :: TOPREALC:6 for c1, c2 being complex number for f, g being complex-valued Function st c1 <> 0 & c2 <> 0 holds (f (/) c1) - (g (/) c2) = ((f (#) c2) - (g (#) c1)) (/) (c1 * c2) proof let c1, c2 be complex number ; ::_thesis: for f, g being complex-valued Function st c1 <> 0 & c2 <> 0 holds (f (/) c1) - (g (/) c2) = ((f (#) c2) - (g (#) c1)) (/) (c1 * c2) let f, g be complex-valued Function; ::_thesis: ( c1 <> 0 & c2 <> 0 implies (f (/) c1) - (g (/) c2) = ((f (#) c2) - (g (#) c1)) (/) (c1 * c2) ) assume A1: ( c1 <> 0 & c2 <> 0 ) ; ::_thesis: (f (/) c1) - (g (/) c2) = ((f (#) c2) - (g (#) c1)) (/) (c1 * c2) A2: dom (f (/) c1) = dom f by VALUED_2:28; A3: dom (g (/) c2) = dom g by VALUED_2:28; A4: dom (f (#) c2) = dom f by VALUED_1:def_5; A5: dom (g (#) c1) = dom g by VALUED_1:def_5; A6: dom ((f (/) c1) - (g (/) c2)) = (dom (f (/) c1)) /\ (dom (g (/) c2)) by VALUED_1:12; A7: dom ((f (#) c2) - (g (#) c1)) = (dom (f (#) c2)) /\ (dom (g (#) c1)) by VALUED_1:12; hence dom ((f (/) c1) - (g (/) c2)) = dom (((f (#) c2) - (g (#) c1)) (/) (c1 * c2)) by A2, A3, A4, A5, A6, VALUED_2:28; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f (/) c1) - (g (/) c2)) or ((f (/) c1) - (g (/) c2)) . b1 = (((f (#) c2) - (g (#) c1)) (/) (c1 * c2)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f (/) c1) - (g (/) c2)) or ((f (/) c1) - (g (/) c2)) . x = (((f (#) c2) - (g (#) c1)) (/) (c1 * c2)) . x ) assume A8: x in dom ((f (/) c1) - (g (/) c2)) ; ::_thesis: ((f (/) c1) - (g (/) c2)) . x = (((f (#) c2) - (g (#) c1)) (/) (c1 * c2)) . x hence ((f (/) c1) - (g (/) c2)) . x = ((f (/) c1) . x) - ((g (/) c2) . x) by VALUED_1:13 .= ((f . x) / c1) - ((g (/) c2) . x) by VALUED_2:29 .= ((f . x) / c1) - ((g . x) / c2) by VALUED_2:29 .= (((f . x) * c2) - ((g . x) * c1)) / (c1 * c2) by A1, XCMPLX_1:130 .= (((f . x) * c2) - ((g (#) c1) . x)) / (c1 * c2) by VALUED_1:6 .= (((f (#) c2) . x) - ((g (#) c1) . x)) / (c1 * c2) by VALUED_1:6 .= (((f (#) c2) - (g (#) c1)) . x) / (c1 * c2) by A2, A3, A4, A5, A6, A7, A8, VALUED_1:13 .= (((f (#) c2) - (g (#) c1)) (/) (c1 * c2)) . x by VALUED_2:29 ; ::_thesis: verum end; theorem :: TOPREALC:7 for c being complex number for f, g being complex-valued Function st c <> 0 holds (f (/) c) - g = (f - (c (#) g)) (/) c proof let c be complex number ; ::_thesis: for f, g being complex-valued Function st c <> 0 holds (f (/) c) - g = (f - (c (#) g)) (/) c let f, g be complex-valued Function; ::_thesis: ( c <> 0 implies (f (/) c) - g = (f - (c (#) g)) (/) c ) assume A1: c <> 0 ; ::_thesis: (f (/) c) - g = (f - (c (#) g)) (/) c A2: dom ((f (/) c) - g) = (dom (f (/) c)) /\ (dom g) by VALUED_1:12 .= (dom f) /\ (dom g) by VALUED_2:28 ; A3: dom (f - (c (#) g)) = (dom f) /\ (dom (c (#) g)) by VALUED_1:12 .= (dom f) /\ (dom g) by VALUED_1:def_5 ; hence dom ((f (/) c) - g) = dom ((f - (c (#) g)) (/) c) by A2, VALUED_2:28; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f (/) c) - g) or ((f (/) c) - g) . b1 = ((f - (c (#) g)) (/) c) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f (/) c) - g) or ((f (/) c) - g) . x = ((f - (c (#) g)) (/) c) . x ) assume A4: x in dom ((f (/) c) - g) ; ::_thesis: ((f (/) c) - g) . x = ((f - (c (#) g)) (/) c) . x hence ((f (/) c) - g) . x = ((f (/) c) . x) - (g . x) by VALUED_1:13 .= ((f . x) / c) - (g . x) by VALUED_2:29 .= ((f . x) / c) - ((c * (g . x)) / c) by A1, XCMPLX_1:89 .= ((f . x) / c) - (((c (#) g) . x) / c) by VALUED_1:6 .= ((f . x) - ((c (#) g) . x)) / c .= ((f - (c (#) g)) . x) / c by A2, A3, A4, VALUED_1:13 .= ((f - (c (#) g)) (/) c) . x by VALUED_2:29 ; ::_thesis: verum end; theorem :: TOPREALC:8 for c, d being complex number for f being complex-valued Function holds (c - d) (#) f = (c (#) f) - (d (#) f) proof let c, d be complex number ; ::_thesis: for f being complex-valued Function holds (c - d) (#) f = (c (#) f) - (d (#) f) let f be complex-valued Function; ::_thesis: (c - d) (#) f = (c (#) f) - (d (#) f) dom ((c (#) f) - (d (#) f)) = (dom (c (#) f)) /\ (dom (d (#) f)) by VALUED_1:12 .= (dom f) /\ (dom (d (#) f)) by VALUED_1:def_5 .= (dom f) /\ (dom f) by VALUED_1:def_5 ; hence A1: dom ((c - d) (#) f) = dom ((c (#) f) - (d (#) f)) by VALUED_1:def_5; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((c - d) (#) f) or ((c - d) (#) f) . b1 = ((c (#) f) - (d (#) f)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((c - d) (#) f) or ((c - d) (#) f) . x = ((c (#) f) - (d (#) f)) . x ) assume A2: x in dom ((c - d) (#) f) ; ::_thesis: ((c - d) (#) f) . x = ((c (#) f) - (d (#) f)) . x thus ((c - d) (#) f) . x = (c - d) * (f . x) by VALUED_1:6 .= (c * (f . x)) - (d * (f . x)) .= (c * (f . x)) - ((d (#) f) . x) by VALUED_1:6 .= ((c (#) f) . x) - ((d (#) f) . x) by VALUED_1:6 .= ((c (#) f) - (d (#) f)) . x by A1, A2, VALUED_1:13 ; ::_thesis: verum end; theorem :: TOPREALC:9 for f, g being complex-valued Function holds (f - g) ^2 = (g - f) ^2 proof let f, g be complex-valued Function; ::_thesis: (f - g) ^2 = (g - f) ^2 A1: dom (f - g) = (dom f) /\ (dom g) by VALUED_1:12; A2: dom (g - f) = (dom g) /\ (dom f) by VALUED_1:12; A3: dom ((f - g) ^2) = dom (f - g) by VALUED_1:11; hence dom ((f - g) ^2) = dom ((g - f) ^2) by A1, A2, VALUED_1:11; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f - g) ^2) or ((f - g) ^2) . b1 = ((g - f) ^2) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f - g) ^2) or ((f - g) ^2) . x = ((g - f) ^2) . x ) assume A4: x in dom ((f - g) ^2) ; ::_thesis: ((f - g) ^2) . x = ((g - f) ^2) . x then A5: (f - g) . x = (f . x) - (g . x) by A3, VALUED_1:13; A6: (g - f) . x = (g . x) - (f . x) by A1, A3, A4, A2, VALUED_1:13; thus ((f - g) ^2) . x = ((f - g) . x) * ((f - g) . x) by VALUED_1:5 .= ((g - f) . x) * ((g - f) . x) by A5, A6 .= ((g - f) ^2) . x by VALUED_1:5 ; ::_thesis: verum end; theorem :: TOPREALC:10 for c being complex number for f being complex-valued Function holds (f (/) c) ^2 = (f ^2) (/) (c ^2) proof let c be complex number ; ::_thesis: for f being complex-valued Function holds (f (/) c) ^2 = (f ^2) (/) (c ^2) let f be complex-valued Function; ::_thesis: (f (/) c) ^2 = (f ^2) (/) (c ^2) thus dom ((f (/) c) ^2) = dom (f (/) c) by VALUED_1:11 .= dom f by VALUED_2:28 .= dom (f ^2) by VALUED_1:11 .= dom ((f ^2) (/) (c ^2)) by VALUED_2:28 ; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f (/) c) ^2) or ((f (/) c) ^2) . b1 = ((f ^2) (/) (c ^2)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f (/) c) ^2) or ((f (/) c) ^2) . x = ((f ^2) (/) (c ^2)) . x ) assume x in dom ((f (/) c) ^2) ; ::_thesis: ((f (/) c) ^2) . x = ((f ^2) (/) (c ^2)) . x thus ((f (/) c) ^2) . x = ((f (/) c) . x) ^2 by VALUED_1:11 .= ((f . x) / c) ^2 by VALUED_2:29 .= ((f . x) ^2) / (c ^2) by XCMPLX_1:76 .= ((f ^2) . x) / (c ^2) by VALUED_1:11 .= ((f ^2) (/) (c ^2)) . x by VALUED_2:29 ; ::_thesis: verum end; theorem Th11: :: TOPREALC:11 for n being Nat for r, s being real number holds |.((n |-> r) - (n |-> s)).| = (sqrt n) * (abs (r - s)) proof let n be Nat; ::_thesis: for r, s being real number holds |.((n |-> r) - (n |-> s)).| = (sqrt n) * (abs (r - s)) let r, s be real number ; ::_thesis: |.((n |-> r) - (n |-> s)).| = (sqrt n) * (abs (r - s)) thus |.((n |-> r) - (n |-> s)).| = sqrt (Sum (sqr (n |-> (r - s)))) by RVSUM_1:30 .= sqrt (Sum (n |-> ((r - s) ^2))) by RVSUM_1:56 .= sqrt (n * ((r - s) ^2)) by RVSUM_1:80 .= (sqrt n) * (sqrt ((r - s) ^2)) by SQUARE_1:29 .= (sqrt n) * (abs (r - s)) by COMPLEX1:72 ; ::_thesis: verum end; registration let f be complex-valued Function; let x be set ; let c be complex number ; clusterf +* (x,c) -> complex-valued ; coherence f +* (x,c) is complex-valued proof set F = f +* (x,c); let a be set ; :: according to VALUED_0:def_7 ::_thesis: ( not a in dom (f +* (x,c)) or (f +* (x,c)) . a is complex ) assume a in dom (f +* (x,c)) ; ::_thesis: (f +* (x,c)) . a is complex percases ( ( x in dom f & x = a ) or ( x in dom f & x <> a ) or not x in dom f ) ; suppose ( x in dom f & x = a ) ; ::_thesis: (f +* (x,c)) . a is complex hence (f +* (x,c)) . a is complex by FUNCT_7:31; ::_thesis: verum end; suppose ( x in dom f & x <> a ) ; ::_thesis: (f +* (x,c)) . a is complex then (f +* (x,c)) . a = f . a by FUNCT_7:32; hence (f +* (x,c)) . a is complex ; ::_thesis: verum end; suppose not x in dom f ; ::_thesis: (f +* (x,c)) . a is complex then (f +* (x,c)) . a = f . a by FUNCT_7:def_3; hence (f +* (x,c)) . a is complex ; ::_thesis: verum end; end; end; end; theorem Th12: :: TOPREALC:12 for x being set for n being Nat for c being complex number holds ((0* n) +* (x,c)) ^2 = (0* n) +* (x,(c ^2)) proof let x be set ; ::_thesis: for n being Nat for c being complex number holds ((0* n) +* (x,c)) ^2 = (0* n) +* (x,(c ^2)) let n be Nat; ::_thesis: for c being complex number holds ((0* n) +* (x,c)) ^2 = (0* n) +* (x,(c ^2)) let c be complex number ; ::_thesis: ((0* n) +* (x,c)) ^2 = (0* n) +* (x,(c ^2)) set f = (0* n) +* (x,c); set g = (0* n) +* (x,(c ^2)); A1: dom ((0* n) +* (x,c)) = dom (0* n) by FUNCT_7:30; A2: dom ((0* n) +* (x,(c ^2))) = dom (0* n) by FUNCT_7:30; A3: dom (((0* n) +* (x,c)) ^2) = dom ((0* n) +* (x,c)) by VALUED_1:11; thus dom (((0* n) +* (x,c)) ^2) = dom ((0* n) +* (x,(c ^2))) by A1, A2, VALUED_1:11; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom (((0* n) +* (x,c)) ^2) or (((0* n) +* (x,c)) ^2) . b1 = ((0* n) +* (x,(c ^2))) . b1 ) let a be set ; ::_thesis: ( not a in dom (((0* n) +* (x,c)) ^2) or (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a ) assume A4: a in dom (((0* n) +* (x,c)) ^2) ; ::_thesis: (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a A5: (((0* n) +* (x,c)) ^2) . a = (((0* n) +* (x,c)) . a) ^2 by VALUED_1:11; percases ( a = x or a <> x ) ; supposeA6: a = x ; ::_thesis: (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a then ((0* n) +* (x,c)) . a = c by A1, A3, A4, FUNCT_7:31; hence (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a by A6, A1, A3, A4, A5, FUNCT_7:31; ::_thesis: verum end; supposeA7: a <> x ; ::_thesis: (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a then A8: ((0* n) +* (x,c)) . a = (n |-> 0) . a by FUNCT_7:32 .= {} . x ; ((0* n) +* (x,(c ^2))) . a = (n |-> 0) . a by A7, FUNCT_7:32 .= {} . x ; hence (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a by A5, A8; ::_thesis: verum end; end; end; theorem Th13: :: TOPREALC:13 for x being set for n being Nat for r being real number st x in Seg n holds |.((0* n) +* (x,r)).| = abs r proof let x be set ; ::_thesis: for n being Nat for r being real number st x in Seg n holds |.((0* n) +* (x,r)).| = abs r let n be Nat; ::_thesis: for r being real number st x in Seg n holds |.((0* n) +* (x,r)).| = abs r let r be real number ; ::_thesis: ( x in Seg n implies |.((0* n) +* (x,r)).| = abs r ) set f = (0* n) +* (x,r); A1: r ^2 in REAL by XREAL_0:def_1; A2: n in NAT by ORDINAL1:def_12; assume A3: x in Seg n ; ::_thesis: |.((0* n) +* (x,r)).| = abs r ((0* n) +* (x,r)) ^2 = (0* n) +* (x,(r ^2)) by Th12; then Sum (((0* n) +* (x,r)) ^2) = r ^2 by A3, A1, A2, JORDAN2B:10; hence |.((0* n) +* (x,r)).| = abs r by COMPLEX1:72; ::_thesis: verum end; theorem Th14: :: TOPREALC:14 for x being set for n being Nat holds (0.REAL n) +* (x,0) = 0.REAL n proof let x be set ; ::_thesis: for n being Nat holds (0.REAL n) +* (x,0) = 0.REAL n let n be Nat; ::_thesis: (0.REAL n) +* (x,0) = 0.REAL n set p = (0.REAL n) +* (x,0); A1: dom ((0.REAL n) +* (x,0)) = Seg n by FINSEQ_1:89; A2: dom ((0.REAL n) +* (x,0)) = dom (0.REAL n) by FUNCT_7:30; A3: dom (0.REAL n) = Seg n by FINSEQ_1:89; now__::_thesis:_for_z_being_set_st_z_in_dom_((0.REAL_n)_+*_(x,0))_holds_ ((0.REAL_n)_+*_(x,0))_._z_=_(0.REAL_n)_._z let z be set ; ::_thesis: ( z in dom ((0.REAL n) +* (x,0)) implies ((0.REAL n) +* (x,0)) . b1 = (0.REAL n) . b1 ) assume A4: z in dom ((0.REAL n) +* (x,0)) ; ::_thesis: ((0.REAL n) +* (x,0)) . b1 = (0.REAL n) . b1 percases ( z = x or z <> x ) ; suppose z = x ; ::_thesis: ((0.REAL n) +* (x,0)) . b1 = (0.REAL n) . b1 hence ((0.REAL n) +* (x,0)) . z = 0 by A1, A3, A4, FUNCT_7:31 .= (0.REAL n) . z ; ::_thesis: verum end; suppose z <> x ; ::_thesis: ((0.REAL n) +* (x,0)) . b1 = (0.REAL n) . b1 hence ((0.REAL n) +* (x,0)) . z = (0.REAL n) . z by FUNCT_7:32; ::_thesis: verum end; end; end; hence (0.REAL n) +* (x,0) = 0.REAL n by A2, FUNCT_1:2; ::_thesis: verum end; theorem Th15: :: TOPREALC:15 for x being set for n being Nat for r being real number for f1 being real-valued b2 -element FinSequence holds mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) proof let x be set ; ::_thesis: for n being Nat for r being real number for f1 being real-valued b1 -element FinSequence holds mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) let n be Nat; ::_thesis: for r being real number for f1 being real-valued n -element FinSequence holds mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) let r be real number ; ::_thesis: for f1 being real-valued n -element FinSequence holds mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) let f1 be real-valued n -element FinSequence; ::_thesis: mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) set p = (0.REAL n) +* (x,r); A1: dom f1 = Seg n by FINSEQ_1:89; A2: dom ((0.REAL n) +* (x,r)) = Seg n by FINSEQ_1:89; A3: dom (mlt (f1,((0.REAL n) +* (x,r)))) = (dom f1) /\ (dom ((0.REAL n) +* (x,r))) by VALUED_1:def_4; A4: dom ((0.REAL n) +* (x,((f1 . x) * r))) = dom (0.REAL n) by FUNCT_7:30; A5: dom (0.REAL n) = Seg n by FINSEQ_1:89; now__::_thesis:_for_z_being_set_st_z_in_dom_(mlt_(f1,((0.REAL_n)_+*_(x,r))))_holds_ (mlt_(f1,((0.REAL_n)_+*_(x,r))))_._z_=_((0.REAL_n)_+*_(x,((f1_._x)_*_r)))_._z let z be set ; ::_thesis: ( z in dom (mlt (f1,((0.REAL n) +* (x,r)))) implies (mlt (f1,((0.REAL n) +* (x,r)))) . b1 = ((0.REAL n) +* (x,((f1 . x) * r))) . b1 ) assume A6: z in dom (mlt (f1,((0.REAL n) +* (x,r)))) ; ::_thesis: (mlt (f1,((0.REAL n) +* (x,r)))) . b1 = ((0.REAL n) +* (x,((f1 . x) * r))) . b1 A7: (mlt (f1,((0.REAL n) +* (x,r)))) . z = (f1 . z) * (((0.REAL n) +* (x,r)) . z) by VALUED_1:5; percases ( z = x or z <> x ) ; supposeA8: z = x ; ::_thesis: (mlt (f1,((0.REAL n) +* (x,r)))) . b1 = ((0.REAL n) +* (x,((f1 . x) * r))) . b1 hence (mlt (f1,((0.REAL n) +* (x,r)))) . z = (f1 . z) * r by A1, A2, A3, A5, A6, A7, FUNCT_7:31 .= ((0.REAL n) +* (x,((f1 . x) * r))) . z by A1, A2, A3, A5, A6, A8, FUNCT_7:31 ; ::_thesis: verum end; supposeA9: z <> x ; ::_thesis: (mlt (f1,((0.REAL n) +* (x,r)))) . b1 = ((0.REAL n) +* (x,((f1 . x) * r))) . b1 hence (mlt (f1,((0.REAL n) +* (x,r)))) . z = (f1 . z) * ((0.REAL n) . z) by A7, FUNCT_7:32 .= (f1 . z) * ((n |-> 0) . z) .= ((0.REAL n) +* (x,((f1 . x) * r))) . z by A9, FUNCT_7:32 ; ::_thesis: verum end; end; end; hence mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) by A4, A5, FINSEQ_1:89, FUNCT_1:2; ::_thesis: verum end; theorem :: TOPREALC:16 for x being set for n being Nat for r being real number for f1 being real-valued b2 -element FinSequence holds |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r proof let x be set ; ::_thesis: for n being Nat for r being real number for f1 being real-valued b1 -element FinSequence holds |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r let n be Nat; ::_thesis: for r being real number for f1 being real-valued n -element FinSequence holds |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r let r be real number ; ::_thesis: for f1 being real-valued n -element FinSequence holds |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r let f1 be real-valued n -element FinSequence; ::_thesis: |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r A1: mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) by Th15; A2: dom f1 = Seg n by FINSEQ_1:89; A3: n in NAT by ORDINAL1:def_12; A4: (f1 . x) * r in REAL by XREAL_0:def_1; percases ( x in dom f1 or not x in dom f1 ) ; suppose x in dom f1 ; ::_thesis: |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r hence |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r by A1, A2, A3, A4, JORDAN2B:10; ::_thesis: verum end; suppose not x in dom f1 ; ::_thesis: |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r then A5: f1 . x = 0 by FUNCT_1:def_2; hence |(f1,((0.REAL n) +* (x,r)))| = Sum (0.REAL n) by A1, Th14 .= (f1 . x) * r by A5, A3, JORDAN2B:9 ; ::_thesis: verum end; end; end; theorem Th17: :: TOPREALC:17 for i, n being Nat for c being complex number for g1 being complex-valued b2 -element FinSequence holds (g1 +* (i,c)) - g1 = (0* n) +* (i,(c - (g1 . i))) proof let i, n be Nat; ::_thesis: for c being complex number for g1 being complex-valued n -element FinSequence holds (g1 +* (i,c)) - g1 = (0* n) +* (i,(c - (g1 . i))) let c be complex number ; ::_thesis: for g1 being complex-valued n -element FinSequence holds (g1 +* (i,c)) - g1 = (0* n) +* (i,(c - (g1 . i))) let g1 be complex-valued n -element FinSequence; ::_thesis: (g1 +* (i,c)) - g1 = (0* n) +* (i,(c - (g1 . i))) A1: dom ((g1 +* (i,c)) - g1) = (dom (g1 +* (i,c))) /\ (dom g1) by VALUED_1:12; A2: dom (0* n) = Seg n by FINSEQ_1:89; A3: dom g1 = Seg n by FINSEQ_1:89; A4: dom (g1 +* (i,c)) = dom g1 by FUNCT_7:30; hence dom ((g1 +* (i,c)) - g1) = dom ((0* n) +* (i,(c - (g1 . i)))) by A1, A3, FINSEQ_1:89; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((g1 +* (i,c)) - g1) or ((g1 +* (i,c)) - g1) . b1 = ((0* n) +* (i,(c - (g1 . i)))) . b1 ) let x be set ; ::_thesis: ( not x in dom ((g1 +* (i,c)) - g1) or ((g1 +* (i,c)) - g1) . x = ((0* n) +* (i,(c - (g1 . i)))) . x ) assume A5: x in dom ((g1 +* (i,c)) - g1) ; ::_thesis: ((g1 +* (i,c)) - g1) . x = ((0* n) +* (i,(c - (g1 . i)))) . x then A6: ((g1 +* (i,c)) - g1) . x = ((g1 +* (i,c)) . x) - (g1 . x) by VALUED_1:13; percases ( x = i or x <> i ) ; supposeA7: x = i ; ::_thesis: ((g1 +* (i,c)) - g1) . x = ((0* n) +* (i,(c - (g1 . i)))) . x hence ((g1 +* (i,c)) - g1) . x = c - (g1 . x) by A6, A1, A5, A4, FUNCT_7:31 .= ((0* n) +* (i,(c - (g1 . i)))) . x by A1, A2, A3, A5, A4, A7, FUNCT_7:31 ; ::_thesis: verum end; supposeA8: x <> i ; ::_thesis: ((g1 +* (i,c)) - g1) . x = ((0* n) +* (i,(c - (g1 . i)))) . x hence ((g1 +* (i,c)) - g1) . x = (g1 . x) - (g1 . x) by A6, FUNCT_7:32 .= (n |-> 0) . x .= ((0* n) +* (i,(c - (g1 . i)))) . x by A8, FUNCT_7:32 ; ::_thesis: verum end; end; end; theorem :: TOPREALC:18 for r being real number holds |.<*r*>.| = abs r proof let r be real number ; ::_thesis: |.<*r*>.| = abs r set f = <*r*>; sqr <*r*> = <*(r ^2)*> by RVSUM_1:55; then Sum (sqr <*r*>) = r ^2 by RVSUM_1:73; hence |.<*r*>.| = abs r by COMPLEX1:72; ::_thesis: verum end; theorem :: TOPREALC:19 for f being real-valued FinSequence holds f is FinSequence of REAL by RVSUM_1:145; theorem :: TOPREALC:20 for f being real-valued FinSequence st |.f.| <> 0 holds ex i being Nat st ( i in dom f & f . i <> 0 ) proof let f be real-valued FinSequence; ::_thesis: ( |.f.| <> 0 implies ex i being Nat st ( i in dom f & f . i <> 0 ) ) assume |.f.| <> 0 ; ::_thesis: ex i being Nat st ( i in dom f & f . i <> 0 ) then consider i being Element of NAT such that A1: i in dom (sqr f) and A2: (sqr f) . i <> 0 by PRVECT_2:3, SQUARE_1:17; take i ; ::_thesis: ( i in dom f & f . i <> 0 ) thus i in dom f by A1, VALUED_1:11; ::_thesis: f . i <> 0 (sqr f) . i = (f . i) ^2 by VALUED_1:11; hence f . i <> 0 by A2; ::_thesis: verum end; theorem Th21: :: TOPREALC:21 for f being real-valued FinSequence holds abs (Sum f) <= Sum (abs f) proof let f be real-valued FinSequence; ::_thesis: abs (Sum f) <= Sum (abs f) defpred S1[ set ] means ex F being real-valued FinSequence st ( F = \$1 & abs (Sum F) <= Sum (abs F) ); A1: S1[ <*> REAL] proof take <*> REAL ; ::_thesis: ( <*> REAL = <*> REAL & abs (Sum (<*> REAL)) <= Sum (abs (<*> REAL)) ) thus ( <*> REAL = <*> REAL & abs (Sum (<*> REAL)) <= Sum (abs (<*> REAL)) ) by ABSVALUE:2, RVSUM_1:72; ::_thesis: verum end; A2: for p being FinSequence of REAL for x being Element of REAL st S1[p] holds S1[p ^ <*x*>] proof let p be FinSequence of REAL ; ::_thesis: for x being Element of REAL st S1[p] holds S1[p ^ <*x*>] let x be Element of REAL ; ::_thesis: ( S1[p] implies S1[p ^ <*x*>] ) given F being real-valued FinSequence such that A3: F = p and A4: abs (Sum F) <= Sum (abs F) ; ::_thesis: S1[p ^ <*x*>] A5: (abs (Sum F)) + (abs x) <= (Sum (abs F)) + (abs x) by A4, XREAL_1:6; take G = p ^ <*x*>; ::_thesis: ( G = p ^ <*x*> & abs (Sum G) <= Sum (abs G) ) A6: abs <*x*> = <*(abs x)*> by JORDAN2B:19; A7: Sum <*(abs x)*> = abs x by RVSUM_1:73; abs G = (abs p) ^ (abs <*x*>) by TOPREAL7:11; then A8: Sum (abs G) = (Sum (abs p)) + (abs x) by A6, A7, RVSUM_1:75; Sum G = (Sum p) + x by RVSUM_1:74; then abs (Sum G) <= (abs (Sum p)) + (abs x) by COMPLEX1:56; hence ( G = p ^ <*x*> & abs (Sum G) <= Sum (abs G) ) by A8, A3, A5, XXREAL_0:2; ::_thesis: verum end; A9: for p being FinSequence of REAL holds S1[p] from FINSEQ_2:sch_2(A1, A2); f is FinSequence of REAL by RVSUM_1:145; then S1[f] by A9; hence abs (Sum f) <= Sum (abs f) ; ::_thesis: verum end; Lm1: for n being Nat for f being real-valued b1 -element FinSequence holds f is Element of n -tuples_on REAL proof let n be Nat; ::_thesis: for f being real-valued n -element FinSequence holds f is Element of n -tuples_on REAL let f be real-valued n -element FinSequence; ::_thesis: f is Element of n -tuples_on REAL A1: f is FinSequence of REAL by RVSUM_1:145; len f = n by CARD_1:def_7; hence f is Element of n -tuples_on REAL by A1, FINSEQ_2:92; ::_thesis: verum end; theorem :: TOPREALC:22 for A being non empty 1-sorted for B being 1 -element 1-sorted for t being Point of B for f being Function of A,B holds f = A --> t proof let A be non empty 1-sorted ; ::_thesis: for B being 1 -element 1-sorted for t being Point of B for f being Function of A,B holds f = A --> t let B be 1 -element 1-sorted ; ::_thesis: for t being Point of B for f being Function of A,B holds f = A --> t let t be Point of B; ::_thesis: for f being Function of A,B holds f = A --> t let f be Function of A,B; ::_thesis: f = A --> t let a be Element of A; :: according to FUNCT_2:def_8 ::_thesis: f . a = (A --> t) . a thus f . a = (A --> t) . a by STRUCT_0:def_10; ::_thesis: verum end; registration let n be non zero Nat; let i be Element of Seg n; let T be non empty real-membered TopSpace; cluster proj (((Seg n) --> T),i) -> real-valued ; coherence proj (((Seg n) --> T),i) is real-valued proof ((Seg n) --> T) . i = T by FUNCOP_1:7; hence proj (((Seg n) --> T),i) is real-valued ; ::_thesis: verum end; end; definition let n be Nat; let p be Element of REAL n; let r be real number ; :: original: (/) redefine funcp (/) r -> Element of REAL n; coherence p (/) r is Element of REAL n proof A1: p (/) r is FinSequence of REAL by RVSUM_1:145; A2: len p = n by CARD_1:def_7; Seg (len (p (/) r)) = dom (p (/) r) by FINSEQ_1:def_3 .= dom p by VALUED_1:def_5 .= Seg n by A2, FINSEQ_1:def_3 ; then len (p (/) r) = n by FINSEQ_1:6; hence p (/) r is Element of REAL n by A1, FINSEQ_2:92; ::_thesis: verum end; end; theorem Th23: :: TOPREALC:23 for m being Nat for r being real number for p, q being Point of (TOP-REAL m) holds ( p in Ball (q,r) iff - p in Ball ((- q),r) ) proof let m be Nat; ::_thesis: for r being real number for p, q being Point of (TOP-REAL m) holds ( p in Ball (q,r) iff - p in Ball ((- q),r) ) let r be real number ; ::_thesis: for p, q being Point of (TOP-REAL m) holds ( p in Ball (q,r) iff - p in Ball ((- q),r) ) let p, q be Point of (TOP-REAL m); ::_thesis: ( p in Ball (q,r) iff - p in Ball ((- q),r) ) A1: m in NAT by ORDINAL1:def_12; A2: now__::_thesis:_for_a,_b_being_Point_of_(TOP-REAL_m)_st_a_in_Ball_(b,r)_holds_ -_a_in_Ball_((-_b),r) let a, b be Point of (TOP-REAL m); ::_thesis: ( a in Ball (b,r) implies - a in Ball ((- b),r) ) assume a in Ball (b,r) ; ::_thesis: - a in Ball ((- b),r) then A3: |.(a - b).| < r by A1, TOPREAL9:7; (- a) - (- b) = (- a) + (- (- b)) .= - (a - b) by RLVECT_1:31 ; then |.((- a) - (- b)).| = |.(a - b).| by EUCLID:71; hence - a in Ball ((- b),r) by A1, A3, TOPREAL9:7; ::_thesis: verum end; ( - (- p) = p & - (- q) = q ) ; hence ( p in Ball (q,r) iff - p in Ball ((- q),r) ) by A2; ::_thesis: verum end; definition let S be 1-sorted ; attrS is complex-functions-membered means :Def1: :: TOPREALC:def 1 the carrier of S is complex-functions-membered ; attrS is real-functions-membered means :Def2: :: TOPREALC:def 2 the carrier of S is real-functions-membered ; end; :: deftheorem Def1 defines complex-functions-membered TOPREALC:def_1_:_ for S being 1-sorted holds ( S is complex-functions-membered iff the carrier of S is complex-functions-membered ); :: deftheorem Def2 defines real-functions-membered TOPREALC:def_2_:_ for S being 1-sorted holds ( S is real-functions-membered iff the carrier of S is real-functions-membered ); registration let n be Nat; cluster TOP-REAL n -> real-functions-membered ; coherence TOP-REAL n is real-functions-membered proof let x be set ; :: according to VALUED_2:def_4,TOPREALC:def_2 ::_thesis: ( not x in the carrier of (TOP-REAL n) or x is set ) assume x in the carrier of (TOP-REAL n) ; ::_thesis: x is set then reconsider x = x as Point of (TOP-REAL n) ; x is real-valued ; hence x is set ; ::_thesis: verum end; end; registration cluster TOP-REAL 0 -> real-membered ; coherence TOP-REAL 0 is real-membered proof thus the carrier of (TOP-REAL 0) is real-membered by EUCLID:22, EUCLID:77; :: according to TOPMETR:def_8 ::_thesis: verum end; end; registration cluster TOP-REAL 0 -> trivial ; coherence TOP-REAL 0 is trivial by EUCLID:22, EUCLID:77; end; registration cluster real-functions-membered -> complex-functions-membered for 1-sorted ; coherence for b1 being 1-sorted st b1 is real-functions-membered holds b1 is complex-functions-membered proof let S be 1-sorted ; ::_thesis: ( S is real-functions-membered implies S is complex-functions-membered ) assume A1: the carrier of S is real-functions-membered ; :: according to TOPREALC:def_2 ::_thesis: S is complex-functions-membered let x be set ; :: according to VALUED_2:def_2,TOPREALC:def_1 ::_thesis: ( not x in the carrier of S or x is set ) thus ( not x in the carrier of S or x is set ) by A1; ::_thesis: verum end; end; registration cluster strict non empty real-functions-membered for 1-sorted ; existence ex b1 being 1-sorted st ( b1 is strict & not b1 is empty & b1 is real-functions-membered ) proof take S = 1-sorted(# { the real-valued Function} #); ::_thesis: ( S is strict & not S is empty & S is real-functions-membered ) thus S is strict ; ::_thesis: ( not S is empty & S is real-functions-membered ) thus not the carrier of S is empty ; :: according to STRUCT_0:def_1 ::_thesis: S is real-functions-membered let x be set ; :: according to VALUED_2:def_4,TOPREALC:def_2 ::_thesis: ( not x in the carrier of S or x is set ) thus ( not x in the carrier of S or x is set ) ; ::_thesis: verum end; end; registration let S be complex-functions-membered 1-sorted ; cluster the carrier of S -> complex-functions-membered ; coherence the carrier of S is complex-functions-membered by Def1; end; registration let S be real-functions-membered 1-sorted ; cluster the carrier of S -> real-functions-membered ; coherence the carrier of S is real-functions-membered by Def2; end; registration cluster non empty strict TopSpace-like real-functions-membered for TopStruct ; existence ex b1 being TopSpace st ( b1 is strict & not b1 is empty & b1 is real-functions-membered ) proof take TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) ; ::_thesis: ( TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is strict & not TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is empty & TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is real-functions-membered ) thus ( TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is strict & not TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is empty & TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is real-functions-membered ) by Def2; ::_thesis: verum end; end; registration let S be complex-functions-membered TopSpace; cluster -> complex-functions-membered for SubSpace of S; coherence for b1 being SubSpace of S holds b1 is complex-functions-membered proof let A be SubSpace of S; ::_thesis: A is complex-functions-membered let x be set ; :: according to VALUED_2:def_2,TOPREALC:def_1 ::_thesis: ( not x in the carrier of A or x is set ) the carrier of A c= the carrier of S by BORSUK_1:1; hence ( not x in the carrier of A or x is set ) ; ::_thesis: verum end; end; registration let S be real-functions-membered TopSpace; cluster -> real-functions-membered for SubSpace of S; coherence for b1 being SubSpace of S holds b1 is real-functions-membered proof let A be SubSpace of S; ::_thesis: A is real-functions-membered let x be set ; :: according to VALUED_2:def_4,TOPREALC:def_2 ::_thesis: ( not x in the carrier of A or x is set ) the carrier of A c= the carrier of S by BORSUK_1:1; hence ( not x in the carrier of A or x is set ) ; ::_thesis: verum end; end; definition let X be complex-functions-membered set ; func (-) X -> complex-functions-membered set means :Def3: :: TOPREALC:def 3 for f being complex-valued Function holds ( - f in it iff f in X ); existence ex b1 being complex-functions-membered set st for f being complex-valued Function holds ( - f in b1 iff f in X ) proof set F = { (- f) where f is Element of X : f in X } ; { (- f) where f is Element of X : f in X } is complex-functions-membered proof let x be set ; :: according to VALUED_2:def_2 ::_thesis: ( not x in { (- f) where f is Element of X : f in X } or x is set ) assume x in { (- f) where f is Element of X : f in X } ; ::_thesis: x is set then ex f being Element of X st ( x = - f & f in X ) ; hence x is set ; ::_thesis: verum end; then reconsider F = { (- f) where f is Element of X : f in X } as complex-functions-membered set ; take F ; ::_thesis: for f being complex-valued Function holds ( - f in F iff f in X ) let f be complex-valued Function; ::_thesis: ( - f in F iff f in X ) hereby ::_thesis: ( f in X implies - f in F ) assume - f in F ; ::_thesis: f in X then ex g being Element of X st ( - f = - g & g in X ) ; hence f in X by RVSUM_1:24; ::_thesis: verum end; thus ( f in X implies - f in F ) ; ::_thesis: verum end; uniqueness for b1, b2 being complex-functions-membered set st ( for f being complex-valued Function holds ( - f in b1 iff f in X ) ) & ( for f being complex-valued Function holds ( - f in b2 iff f in X ) ) holds b1 = b2 proof let A, B be complex-functions-membered set ; ::_thesis: ( ( for f being complex-valued Function holds ( - f in A iff f in X ) ) & ( for f being complex-valued Function holds ( - f in B iff f in X ) ) implies A = B ) assume that A1: for f being complex-valued Function holds ( - f in A iff f in X ) and A2: for f being complex-valued Function holds ( - f in B iff f in X ) ; ::_thesis: A = B thus A c= B :: according to XBOOLE_0:def_10 ::_thesis: B c= A proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in B ) assume A3: x in A ; ::_thesis: x in B then reconsider x = x as Element of A ; A4: - (- x) = x ; then - x in X by A1, A3; hence x in B by A2, A4; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in B or x in A ) assume A5: x in B ; ::_thesis: x in A then reconsider x = x as Element of B ; A6: - (- x) = x ; then - x in X by A2, A5; hence x in A by A1, A6; ::_thesis: verum end; involutiveness for b1, b2 being complex-functions-membered set st ( for f being complex-valued Function holds ( - f in b1 iff f in b2 ) ) holds for f being complex-valued Function holds ( - f in b2 iff f in b1 ) proof let r, A be complex-functions-membered set ; ::_thesis: ( ( for f being complex-valued Function holds ( - f in r iff f in A ) ) implies for f being complex-valued Function holds ( - f in A iff f in r ) ) assume A7: for f being complex-valued Function holds ( - f in r iff f in A ) ; ::_thesis: for f being complex-valued Function holds ( - f in A iff f in r ) let f be complex-valued Function; ::_thesis: ( - f in A iff f in r ) hereby ::_thesis: ( f in r implies - f in A ) assume - f in A ; ::_thesis: f in r then - (- f) in r by A7; hence f in r ; ::_thesis: verum end; assume f in r ; ::_thesis: - f in A then - (- f) in r ; hence - f in A by A7; ::_thesis: verum end; end; :: deftheorem Def3 defines (-) TOPREALC:def_3_:_ for X, b2 being complex-functions-membered set holds ( b2 = (-) X iff for f being complex-valued Function holds ( - f in b2 iff f in X ) ); registration let X be empty set ; cluster (-) X -> empty complex-functions-membered ; coherence (-) X is empty proof assume not (-) X is empty ; ::_thesis: contradiction then consider x being set such that A1: x in (-) X by XBOOLE_0:def_1; reconsider x = x as Element of (-) X by A1; - (- x) = x ; hence contradiction by A1, Def3; ::_thesis: verum end; end; registration let X be non empty complex-functions-membered set ; cluster (-) X -> non empty complex-functions-membered ; coherence not (-) X is empty proof consider x being set such that A1: x in X by XBOOLE_0:def_1; reconsider x = x as Element of X by A1; - x in (-) X by Def3; hence not (-) X is empty ; ::_thesis: verum end; end; theorem Th24: :: TOPREALC:24 for X being complex-functions-membered set for f being complex-valued Function holds ( - f in X iff f in (-) X ) proof let X be complex-functions-membered set ; ::_thesis: for f being complex-valued Function holds ( - f in X iff f in (-) X ) let f be complex-valued Function; ::_thesis: ( - f in X iff f in (-) X ) ( - (- f) = f & (-) ((-) X) = X ) ; hence ( - f in X iff f in (-) X ) by Def3; ::_thesis: verum end; registration let X be real-functions-membered set ; cluster (-) X -> complex-functions-membered real-functions-membered ; coherence (-) X is real-functions-membered proof let x be set ; :: according to VALUED_2:def_4 ::_thesis: ( not x in (-) X or x is set ) assume A1: x in (-) X ; ::_thesis: x is set then reconsider x = x as Element of (-) X ; A2: - x in X by A1, Th24; ( - (- x) = x & (-) ((-) X) = X ) ; hence x is set by A2; ::_thesis: verum end; end; theorem Th25: :: TOPREALC:25 for n being Nat for X being Subset of (TOP-REAL n) holds - X = (-) X proof let n be Nat; ::_thesis: for X being Subset of (TOP-REAL n) holds - X = (-) X set T = TOP-REAL n; let X be Subset of (TOP-REAL n); ::_thesis: - X = (-) X for f being complex-valued Function holds ( - f in - X iff f in X ) proof let f be complex-valued Function; ::_thesis: ( - f in - X iff f in X ) hereby ::_thesis: ( f in X implies - f in - X ) assume - f in - X ; ::_thesis: f in X then consider v being Element of (TOP-REAL n) such that A1: - f = (- 1) * v and A2: v in X ; reconsider g = - f as Element of (TOP-REAL n) by A1; - g = - (- v) by A1 .= v ; hence f in X by A2; ::_thesis: verum end; assume A3: f in X ; ::_thesis: - f in - X then reconsider g = f as Element of (TOP-REAL n) ; - g = (- 1) * g ; hence - f in - X by A3; ::_thesis: verum end; hence - X = (-) X by Def3; ::_thesis: verum end; definition let n be Nat; let X be Subset of (TOP-REAL n); :: original: (-) redefine func (-) X -> Subset of (TOP-REAL n); coherence (-) X is Subset of (TOP-REAL n) proof - X = (-) X by Th25; hence (-) X is Subset of (TOP-REAL n) ; ::_thesis: verum end; end; registration let n be Nat; let X be open Subset of (TOP-REAL n); cluster (-) X -> open for Subset of (TOP-REAL n); coherence for b1 being Subset of (TOP-REAL n) st b1 = (-) X holds b1 is open proof reconsider n = n as Element of NAT by ORDINAL1:def_12; reconsider Y = (-) X as Subset of (TOP-REAL n) ; for p being Point of (TOP-REAL n) st p in Y holds ex r being positive real number st Ball (p,r) c= Y proof let p be Point of (TOP-REAL n); ::_thesis: ( p in Y implies ex r being positive real number st Ball (p,r) c= Y ) assume p in Y ; ::_thesis: ex r being positive real number st Ball (p,r) c= Y then - p in (-) Y by Def3; then consider r being positive real number such that A1: Ball ((- p),r) c= X by TOPS_4:2; take r ; ::_thesis: Ball (p,r) c= Y let x be Element of (TOP-REAL n); :: according to LATTICE7:def_1 ::_thesis: ( not x in Ball (p,r) or x in Y ) assume x in Ball (p,r) ; ::_thesis: x in Y then - x in Ball ((- p),r) by Th23; hence x in Y by A1, Th24; ::_thesis: verum end; hence for b1 being Subset of (TOP-REAL n) st b1 = (-) X holds b1 is open by TOPS_4:2; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let x be set ; :: original: . redefine funcp . x -> Element of REAL ; coherence p . x is Element of REAL by XREAL_0:def_1; end; definition let R, S, T be non empty TopSpace; let f be Function of [:R,S:],T; let x be Point of [:R,S:]; :: original: . redefine funcf . x -> Point of T; coherence f . x is Point of T proof consider x1 being Point of R, x2 being Point of S such that A1: x = [x1,x2] by BORSUK_1:10; f . [x1,x2] is Point of T ; hence f . x is Point of T by A1; ::_thesis: verum end; end; definition let R, S, T be non empty TopSpace; let f be Function of [:R,S:],T; let r be Point of R; let s be Point of S; :: original: . redefine funcf . (r,s) -> Point of T; coherence f . (r,s) is Point of T proof f . (r,s) = f . [r,s] ; hence f . (r,s) is Point of T ; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let r be real number ; :: original: + redefine funcp + r -> Point of (TOP-REAL n); coherence r + p is Point of (TOP-REAL n) proof A1: p + r is FinSequence of REAL by RVSUM_1:145; A2: len p = n by CARD_1:def_7; Seg (len (p + r)) = dom (p + r) by FINSEQ_1:def_3 .= dom p by VALUED_1:def_2 .= Seg n by A2, FINSEQ_1:def_3 ; then len (p + r) = n by FINSEQ_1:6; then p + r is Element of REAL n by A1, FINSEQ_2:92; hence r + p is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let r be real number ; :: original: - redefine funcp - r -> Point of (TOP-REAL n); coherence p - r is Point of (TOP-REAL n) proof p + (- r) is Point of (TOP-REAL n) ; hence p - r is Point of (TOP-REAL n) ; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let r be real number ; :: original: (#) redefine funcp (#) r -> Point of (TOP-REAL n); coherence r (#) p is Point of (TOP-REAL n) proof A1: p (#) r is FinSequence of REAL by RVSUM_1:145; A2: len p = n by CARD_1:def_7; Seg (len (p (#) r)) = dom (p (#) r) by FINSEQ_1:def_3 .= dom p by VALUED_1:def_5 .= Seg n by A2, FINSEQ_1:def_3 ; then len (p (#) r) = n by FINSEQ_1:6; then p (#) r is Element of REAL n by A1, FINSEQ_2:92; hence r (#) p is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let r be real number ; :: original: (/) redefine funcp (/) r -> Point of (TOP-REAL n); coherence p (/) r is Point of (TOP-REAL n) proof reconsider f = p as Element of REAL n by EUCLID:22; f (/) r is Element of REAL n ; hence p (/) r is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; end; definition let n be Nat; let p1, p2 be Point of (TOP-REAL n); :: original: (#) redefine funcp1 (#) p2 -> Point of (TOP-REAL n); coherence p1 (#) p2 is Point of (TOP-REAL n) proof A1: p1 (#) p2 is FinSequence of REAL by RVSUM_1:145; A2: ( len p1 = n & len p2 = n ) by CARD_1:def_7; Seg (len (p1 (#) p2)) = dom (p1 (#) p2) by FINSEQ_1:def_3 .= (dom p1) /\ (dom p2) by VALUED_1:def_4 .= (Seg n) /\ (dom p2) by A2, FINSEQ_1:def_3 .= (Seg n) /\ (Seg n) by A2, FINSEQ_1:def_3 ; then len (p1 (#) p2) = n by FINSEQ_1:6; then p1 (#) p2 is Element of REAL n by A1, FINSEQ_2:92; hence p1 (#) p2 is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; commutativity for p1, p2 being Point of (TOP-REAL n) holds p1 (#) p2 = p2 (#) p1 ; end; definition let n be Nat; let p be Point of (TOP-REAL n); :: original: ^2 redefine func sqr p -> Point of (TOP-REAL n); coherence p ^2 is Point of (TOP-REAL n) proof sqr p = p (#) p ; hence p ^2 is Point of (TOP-REAL n) ; ::_thesis: verum end; end; definition let n be Nat; let p1, p2 be Point of (TOP-REAL n); :: original: /" redefine funcp1 /" p2 -> Point of (TOP-REAL n); coherence p1 /" p2 is Point of (TOP-REAL n) proof A1: p1 /" p2 is FinSequence of REAL by RVSUM_1:145; A2: ( len p1 = n & len p2 = n ) by CARD_1:def_7; Seg (len (p1 /" p2)) = dom (p1 /" p2) by FINSEQ_1:def_3 .= (dom p1) /\ (dom p2) by VALUED_1:16 .= (Seg n) /\ (dom p2) by A2, FINSEQ_1:def_3 .= (Seg n) /\ (Seg n) by A2, FINSEQ_1:def_3 ; then len (p1 /" p2) = n by FINSEQ_1:6; then p1 /" p2 is Element of REAL n by A1, FINSEQ_2:92; hence p1 /" p2 is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let x be set ; let r be real number ; :: original: +* redefine funcp +* (x,r) -> Point of (TOP-REAL n); coherence p +* (x,r) is Point of (TOP-REAL n) proof A1: p +* (x,r) is FinSequence of REAL by RVSUM_1:145; A2: len p = n by CARD_1:def_7; Seg (len (p +* (x,r))) = dom (p +* (x,r)) by FINSEQ_1:def_3 .= dom p by FUNCT_7:30 .= Seg n by A2, FINSEQ_1:def_3 ; then len (p +* (x,r)) = n by FINSEQ_1:6; then p +* (x,r) is Element of REAL n by A1, FINSEQ_2:92; hence p +* (x,r) is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; end; theorem :: TOPREALC:26 for n being Nat for r being real number for a, o being Point of (TOP-REAL n) st n <> 0 & a in Ball (o,r) holds abs (Sum (a - o)) < n * r proof let n be Nat; ::_thesis: for r being real number for a, o being Point of (TOP-REAL n) st n <> 0 & a in Ball (o,r) holds abs (Sum (a - o)) < n * r let r be real number ; ::_thesis: for a, o being Point of (TOP-REAL n) st n <> 0 & a in Ball (o,r) holds abs (Sum (a - o)) < n * r let a, o be Point of (TOP-REAL n); ::_thesis: ( n <> 0 & a in Ball (o,r) implies abs (Sum (a - o)) < n * r ) set R1 = a - o; set R2 = n |-> r; assume that A1: n <> 0 and A2: a in Ball (o,r) ; ::_thesis: abs (Sum (a - o)) < n * r A3: Sum (n |-> r) = n * r by RVSUM_1:80; A4: ( abs (a - o) is Element of n -tuples_on REAL & n |-> r is Element of n -tuples_on REAL ) by Lm1; A5: for j being Nat st j in Seg n holds (abs (a - o)) . j < (n |-> r) . j proof let j be Nat; ::_thesis: ( j in Seg n implies (abs (a - o)) . j < (n |-> r) . j ) assume j in Seg n ; ::_thesis: (abs (a - o)) . j < (n |-> r) . j then A6: (n |-> r) . j = r by FINSEQ_2:57; abs ((a - o) . j) < r by A2, EUCLID_9:9; hence (abs (a - o)) . j < (n |-> r) . j by A6, VALUED_1:18; ::_thesis: verum end; then A7: for j being Nat st j in Seg n holds (abs (a - o)) . j <= (n |-> r) . j ; ex j being Nat st ( j in Seg n & (abs (a - o)) . j < (n |-> r) . j ) proof take 1 ; ::_thesis: ( 1 in Seg n & (abs (a - o)) . 1 < (n |-> r) . 1 ) 1 <= n by A1, NAT_1:14; hence 1 in Seg n ; ::_thesis: (abs (a - o)) . 1 < (n |-> r) . 1 hence (abs (a - o)) . 1 < (n |-> r) . 1 by A5; ::_thesis: verum end; then A8: Sum (abs (a - o)) < Sum (n |-> r) by A7, A4, RVSUM_1:83; abs (Sum (a - o)) <= Sum (abs (a - o)) by Th21; hence abs (Sum (a - o)) < n * r by A3, A8, XXREAL_0:2; ::_thesis: verum end; registration let n be Nat; cluster Euclid n -> real-functions-membered ; coherence Euclid n is real-functions-membered proof let x be set ; :: according to VALUED_2:def_4,TOPREALC:def_2 ::_thesis: ( not x in the carrier of (Euclid n) or x is set ) assume x in the carrier of (Euclid n) ; ::_thesis: x is set then reconsider x = x as Point of (Euclid n) ; x is real-valued ; hence x is set ; ::_thesis: verum end; end; theorem :: TOPREALC:27 for V being non empty right_complementable add-associative right_zeroed addLoopStr for v, u being Element of V holds (v + u) - u = v proof let V be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for v, u being Element of V holds (v + u) - u = v let v, u be Element of V; ::_thesis: (v + u) - u = v thus (v + u) - u = v + (u - u) by RLVECT_1:28 .= v + (0. V) by RLVECT_1:5 .= v by RLVECT_1:4 ; ::_thesis: verum end; theorem :: TOPREALC:28 for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr for v, u being Element of V holds (v - u) + u = v proof let V be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for v, u being Element of V holds (v - u) + u = v let v, u be Element of V; ::_thesis: (v - u) + u = v thus (v - u) + u = v - (u - u) by RLVECT_1:29 .= v - (0. V) by RLVECT_1:5 .= v by RLVECT_1:13 ; ::_thesis: verum end; theorem Th29: :: TOPREALC:29 for X being set for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [+] c = f <+> ((dom f) --> c) proof let X be set ; ::_thesis: for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [+] c = f <+> ((dom f) --> c) let c be complex number ; ::_thesis: for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [+] c = f <+> ((dom f) --> c) let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y holds f [+] c = f <+> ((dom f) --> c) let f be PartFunc of X,Y; ::_thesis: f [+] c = f <+> ((dom f) --> c) set g = (dom f) --> c; A1: dom (f [+] c) = dom f by VALUED_2:def_37; A2: dom (f <+> ((dom f) --> c)) = (dom f) /\ (dom ((dom f) --> c)) by VALUED_2:def_41; A3: dom ((dom f) --> c) = dom f by FUNCOP_1:13; now__::_thesis:_for_x_being_set_st_x_in_dom_(f_[+]_c)_holds_ (f_[+]_c)_._x_=_(f_<+>_((dom_f)_-->_c))_._x let x be set ; ::_thesis: ( x in dom (f [+] c) implies (f [+] c) . x = (f <+> ((dom f) --> c)) . x ) assume A4: x in dom (f [+] c) ; ::_thesis: (f [+] c) . x = (f <+> ((dom f) --> c)) . x hence (f [+] c) . x = (f . x) + c by VALUED_2:def_37 .= (f . x) + (((dom f) --> c) . x) by A1, A4, FUNCOP_1:7 .= (f <+> ((dom f) --> c)) . x by A1, A2, A3, A4, VALUED_2:def_41 ; ::_thesis: verum end; hence f [+] c = f <+> ((dom f) --> c) by A1, A2, A3, FUNCT_1:2; ::_thesis: verum end; theorem :: TOPREALC:30 for X being set for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [-] c = f <-> ((dom f) --> c) proof let X be set ; ::_thesis: for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [-] c = f <-> ((dom f) --> c) let c be complex number ; ::_thesis: for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [-] c = f <-> ((dom f) --> c) let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y holds f [-] c = f <-> ((dom f) --> c) let f be PartFunc of X,Y; ::_thesis: f [-] c = f <-> ((dom f) --> c) set g = (dom f) --> c; A1: dom (f [-] c) = dom f by VALUED_2:def_37; A2: dom (f <-> ((dom f) --> c)) = (dom f) /\ (dom ((dom f) --> c)) by VALUED_2:61; A3: dom ((dom f) --> c) = dom f by FUNCOP_1:13; now__::_thesis:_for_x_being_set_st_x_in_dom_(f_[-]_c)_holds_ (f_[-]_c)_._x_=_(f_<->_((dom_f)_-->_c))_._x let x be set ; ::_thesis: ( x in dom (f [-] c) implies (f [-] c) . x = (f <-> ((dom f) --> c)) . x ) assume A4: x in dom (f [-] c) ; ::_thesis: (f [-] c) . x = (f <-> ((dom f) --> c)) . x hence (f [-] c) . x = (f . x) - c by VALUED_2:def_37 .= (f . x) - (((dom f) --> c) . x) by A1, A4, FUNCOP_1:7 .= (f <-> ((dom f) --> c)) . x by A1, A2, A3, A4, VALUED_2:62 ; ::_thesis: verum end; hence f [-] c = f <-> ((dom f) --> c) by A1, A2, A3, FUNCT_1:2; ::_thesis: verum end; theorem Th31: :: TOPREALC:31 for X being set for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [#] c = f <#> ((dom f) --> c) proof let X be set ; ::_thesis: for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [#] c = f <#> ((dom f) --> c) let c be complex number ; ::_thesis: for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [#] c = f <#> ((dom f) --> c) let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y holds f [#] c = f <#> ((dom f) --> c) let f be PartFunc of X,Y; ::_thesis: f [#] c = f <#> ((dom f) --> c) set g = (dom f) --> c; A1: dom (f [#] c) = dom f by VALUED_2:def_39; A2: dom (f <#> ((dom f) --> c)) = (dom f) /\ (dom ((dom f) --> c)) by VALUED_2:def_43; A3: dom ((dom f) --> c) = dom f by FUNCOP_1:13; now__::_thesis:_for_x_being_set_st_x_in_dom_(f_[#]_c)_holds_ (f_[#]_c)_._x_=_(f_<#>_((dom_f)_-->_c))_._x let x be set ; ::_thesis: ( x in dom (f [#] c) implies (f [#] c) . x = (f <#> ((dom f) --> c)) . x ) assume A4: x in dom (f [#] c) ; ::_thesis: (f [#] c) . x = (f <#> ((dom f) --> c)) . x hence (f [#] c) . x = c (#) (f . x) by VALUED_2:def_39 .= (f . x) (#) (((dom f) --> c) . x) by A1, A4, FUNCOP_1:7 .= (f <#> ((dom f) --> c)) . x by A1, A2, A3, A4, VALUED_2:def_43 ; ::_thesis: verum end; hence f [#] c = f <#> ((dom f) --> c) by A1, A2, A3, FUNCT_1:2; ::_thesis: verum end; theorem :: TOPREALC:32 for X being set for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [/] c = f ((dom f) --> c) proof let X be set ; ::_thesis: for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [/] c = f ((dom f) --> c) let c be complex number ; ::_thesis: for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [/] c = f ((dom f) --> c) let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y holds f [/] c = f ((dom f) --> c) let f be PartFunc of X,Y; ::_thesis: f [/] c = f ((dom f) --> c) set g = (dom f) --> c; A1: dom (f [/] c) = dom f by VALUED_2:def_39; A2: dom (f ((dom f) --> c)) = (dom f) /\ (dom ((dom f) --> c)) by VALUED_2:71; A3: dom ((dom f) --> c) = dom f by FUNCOP_1:13; now__::_thesis:_for_x_being_set_st_x_in_dom_(f_[/]_c)_holds_ (f_[/]_c)_._x_=_(f__((dom_f)_-->_c))_._x let x be set ; ::_thesis: ( x in dom (f [/] c) implies (f [/] c) . x = (f ((dom f) --> c)) . x ) assume A4: x in dom (f [/] c) ; ::_thesis: (f [/] c) . x = (f ((dom f) --> c)) . x hence (f [/] c) . x = (f . x) (/) c by VALUED_2:56 .= (f . x) (/) (((dom f) --> c) . x) by A1, A4, FUNCOP_1:7 .= (f ((dom f) --> c)) . x by A1, A2, A3, A4, VALUED_2:72 ; ::_thesis: verum end; hence f [/] c = f ((dom f) --> c) by A1, A2, A3, FUNCT_1:2; ::_thesis: verum end; registration let D be complex-functions-membered set ; let f, g be FinSequence of D; clusterf <++> g -> FinSequence-like ; coherence f <++> g is FinSequence-like proof dom (f <++> g) = (dom f) /\ (dom g) by VALUED_2:def_45; hence f <++> g is FinSequence-like by VALUED_1:19; ::_thesis: verum end; clusterf <--> g -> FinSequence-like ; coherence f <--> g is FinSequence-like proof dom (f <--> g) = (dom f) /\ (dom g) by VALUED_2:def_46; hence f <--> g is FinSequence-like by VALUED_1:19; ::_thesis: verum end; clusterf <##> g -> FinSequence-like ; coherence f <##> g is FinSequence-like proof dom (f <##> g) = (dom f) /\ (dom g) by VALUED_2:def_47; hence f <##> g is FinSequence-like by VALUED_1:19; ::_thesis: verum end; clusterf g -> FinSequence-like ; coherence f g is FinSequence-like proof dom (f g) = (dom f) /\ (dom g) by VALUED_2:def_48; hence f g is FinSequence-like by VALUED_1:19; ::_thesis: verum end; end; theorem :: TOPREALC:33 for X being set for n being Nat for f being Function of X,(TOP-REAL n) holds <-> f is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f being Function of X,(TOP-REAL n) holds <-> f is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f being Function of X,(TOP-REAL n) holds <-> f is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: <-> f is Function of X,(TOP-REAL n) set g = <-> f; A1: dom (<-> f) = dom f by VALUED_2:def_33; A2: dom f = X by FUNCT_2:def_1; for x being set st x in X holds (<-> f) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (<-> f) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (<-> f) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f as Function of X,(TOP-REAL n) ; A4: - (f . x) = - (f . x) ; (<-> f) . x = - (f . x) by A1, A2, VALUED_2:def_33; hence (<-> f) . x in the carrier of (TOP-REAL n) by A4; ::_thesis: verum end; hence <-> f is Function of X,(TOP-REAL n) by A1, A2, FUNCT_2:3; ::_thesis: verum end; theorem Th34: :: TOPREALC:34 for i, n being Nat for f being Function of (TOP-REAL i),(TOP-REAL n) holds f (-) is Function of (TOP-REAL i),(TOP-REAL n) proof let i, n be Nat; ::_thesis: for f being Function of (TOP-REAL i),(TOP-REAL n) holds f (-) is Function of (TOP-REAL i),(TOP-REAL n) set X = the carrier of (TOP-REAL i); let f be Function of the carrier of (TOP-REAL i),(TOP-REAL n); ::_thesis: f (-) is Function of (TOP-REAL i),(TOP-REAL n) set g = f (-) ; A1: dom (f (-)) = dom f by VALUED_2:def_34; A2: dom f = the carrier of (TOP-REAL i) by FUNCT_2:def_1; for x being set st x in the carrier of (TOP-REAL i) holds (f (-)) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in the carrier of (TOP-REAL i) implies (f (-)) . x in the carrier of (TOP-REAL n) ) assume x in the carrier of (TOP-REAL i) ; ::_thesis: (f (-)) . x in the carrier of (TOP-REAL n) then reconsider x = x as Element of the carrier of (TOP-REAL i) ; reconsider y = - x as Element of the carrier of (TOP-REAL i) ; reconsider f = f as Function of the carrier of (TOP-REAL i),(TOP-REAL n) ; (f (-)) . x = f . y by A1, A2, VALUED_2:def_34; hence (f (-)) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f (-) is Function of (TOP-REAL i),(TOP-REAL n) by A1, A2, FUNCT_2:3; ::_thesis: verum end; theorem Th35: :: TOPREALC:35 for X being set for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [+] r is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [+] r is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for r being real number for f being Function of X,(TOP-REAL n) holds f [+] r is Function of X,(TOP-REAL n) let r be real number ; ::_thesis: for f being Function of X,(TOP-REAL n) holds f [+] r is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: f [+] r is Function of X,(TOP-REAL n) set h = f [+] r; dom f = X by FUNCT_2:def_1; then A1: dom (f [+] r) = X by VALUED_2:def_37; for x being set st x in X holds (f [+] r) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f [+] r) . x in the carrier of (TOP-REAL n) ) assume A2: x in X ; ::_thesis: (f [+] r) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A2; reconsider f = f as Function of X,(TOP-REAL n) ; A3: (f . x) + r = (f . x) + r ; (f [+] r) . x = (f . x) + r by A1, VALUED_2:def_37; hence (f [+] r) . x in the carrier of (TOP-REAL n) by A3; ::_thesis: verum end; hence f [+] r is Function of X,(TOP-REAL n) by A1, FUNCT_2:3; ::_thesis: verum end; theorem :: TOPREALC:36 for X being set for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [-] r is Function of X,(TOP-REAL n) by Th35; theorem Th37: :: TOPREALC:37 for X being set for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [#] r is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [#] r is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for r being real number for f being Function of X,(TOP-REAL n) holds f [#] r is Function of X,(TOP-REAL n) let r be real number ; ::_thesis: for f being Function of X,(TOP-REAL n) holds f [#] r is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: f [#] r is Function of X,(TOP-REAL n) set h = f [#] r; dom f = X by FUNCT_2:def_1; then A1: dom (f [#] r) = X by VALUED_2:def_39; for x being set st x in X holds (f [#] r) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f [#] r) . x in the carrier of (TOP-REAL n) ) assume A2: x in X ; ::_thesis: (f [#] r) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A2; reconsider f = f as Function of X,(TOP-REAL n) ; A3: (f . x) (#) r = (f . x) (#) r ; (f [#] r) . x = (f . x) (#) r by A1, VALUED_2:def_39; hence (f [#] r) . x in the carrier of (TOP-REAL n) by A3; ::_thesis: verum end; hence f [#] r is Function of X,(TOP-REAL n) by A1, FUNCT_2:3; ::_thesis: verum end; theorem :: TOPREALC:38 for X being set for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [/] r is Function of X,(TOP-REAL n) by Th37; theorem Th39: :: TOPREALC:39 for X being set for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <++> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <++> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f, g being Function of X,(TOP-REAL n) holds f <++> g is Function of X,(TOP-REAL n) let f, g be Function of X,(TOP-REAL n); ::_thesis: f <++> g is Function of X,(TOP-REAL n) set h = f <++> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <++> g) = X by A1, VALUED_2:def_45; for x being set st x in X holds (f <++> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <++> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <++> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f, g = g as Function of X,(TOP-REAL n) ; A4: (f . x) + (g . x) = (f . x) + (g . x) ; (f <++> g) . x = (f . x) + (g . x) by A2, VALUED_2:def_45; hence (f <++> g) . x in the carrier of (TOP-REAL n) by A4; ::_thesis: verum end; hence f <++> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th40: :: TOPREALC:40 for X being set for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <--> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <--> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f, g being Function of X,(TOP-REAL n) holds f <--> g is Function of X,(TOP-REAL n) let f, g be Function of X,(TOP-REAL n); ::_thesis: f <--> g is Function of X,(TOP-REAL n) set h = f <--> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <--> g) = X by A1, VALUED_2:def_46; for x being set st x in X holds (f <--> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <--> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <--> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f, g = g as Function of X,(TOP-REAL n) ; A4: (f . x) - (g . x) = (f . x) - (g . x) ; (f <--> g) . x = (f . x) - (g . x) by A2, VALUED_2:def_46; hence (f <--> g) . x in the carrier of (TOP-REAL n) by A4; ::_thesis: verum end; hence f <--> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th41: :: TOPREALC:41 for X being set for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <##> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <##> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f, g being Function of X,(TOP-REAL n) holds f <##> g is Function of X,(TOP-REAL n) let f, g be Function of X,(TOP-REAL n); ::_thesis: f <##> g is Function of X,(TOP-REAL n) set h = f <##> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <##> g) = X by A1, VALUED_2:def_47; for x being set st x in X holds (f <##> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <##> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <##> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f, g = g as Function of X,(TOP-REAL n) ; (f <##> g) . x = (f . x) (#) (g . x) by A2, VALUED_2:def_47; hence (f <##> g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f <##> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th42: :: TOPREALC:42 for X being set for n being Nat for f, g being Function of X,(TOP-REAL n) holds f g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f, g being Function of X,(TOP-REAL n) holds f g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f, g being Function of X,(TOP-REAL n) holds f g is Function of X,(TOP-REAL n) let f, g be Function of X,(TOP-REAL n); ::_thesis: f g is Function of X,(TOP-REAL n) set h = f g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f g) = X by A1, VALUED_2:def_48; for x being set st x in X holds (f g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f, g = g as Function of X,(TOP-REAL n) ; (f g) . x = (f . x) /" (g . x) by A2, VALUED_2:def_48; hence (f g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th43: :: TOPREALC:43 for X being set for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <+> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <+> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <+> g is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: for g being Function of X,R^1 holds f <+> g is Function of X,(TOP-REAL n) let g be Function of X,R^1; ::_thesis: f <+> g is Function of X,(TOP-REAL n) set h = f <+> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <+> g) = X by A1, VALUED_2:def_41; for x being set st x in X holds (f <+> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <+> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <+> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f as Function of X,(TOP-REAL n) ; (f <+> g) . x = (f . x) + (g . x) by A2, VALUED_2:def_41; hence (f <+> g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f <+> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th44: :: TOPREALC:44 for X being set for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <-> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <-> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <-> g is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: for g being Function of X,R^1 holds f <-> g is Function of X,(TOP-REAL n) let g be Function of X,R^1; ::_thesis: f <-> g is Function of X,(TOP-REAL n) set h = f <-> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <-> g) = X by A1, VALUED_2:61; for x being set st x in X holds (f <-> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <-> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <-> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f as Function of X,(TOP-REAL n) ; (f <-> g) . x = (f . x) - (g . x) by A2, VALUED_2:62; hence (f <-> g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f <-> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th45: :: TOPREALC:45 for X being set for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <#> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <#> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <#> g is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: for g being Function of X,R^1 holds f <#> g is Function of X,(TOP-REAL n) let g be Function of X,R^1; ::_thesis: f <#> g is Function of X,(TOP-REAL n) set h = f <#> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <#> g) = X by A1, VALUED_2:def_43; for x being set st x in X holds (f <#> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <#> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <#> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f as Function of X,(TOP-REAL n) ; (f <#> g) . x = (f . x) (#) (g . x) by A2, VALUED_2:def_43; hence (f <#> g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f <#> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th46: :: TOPREALC:46 for X being set for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f g is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: for g being Function of X,R^1 holds f g is Function of X,(TOP-REAL n) let g be Function of X,R^1; ::_thesis: f g is Function of X,(TOP-REAL n) set h = f g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f g) = X by A1, VALUED_2:71; for x being set st x in X holds (f g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f as Function of X,(TOP-REAL n) ; (f g) . x = (f . x) (/) (g . x) by A2, VALUED_2:72; hence (f g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; definition let n be Nat; let T be non empty set ; let R be real-membered set ; let f be Function of T,R; func incl (f,n) -> Function of T,(TOP-REAL n) means :Def4: :: TOPREALC:def 4 for t being Element of T holds it . t = n |-> (f . t); existence ex b1 being Function of T,(TOP-REAL n) st for t being Element of T holds b1 . t = n |-> (f . t) proof defpred S1[ set , set ] means \$2 = n |-> (f . \$1); A1: for x being Element of T ex y being Element of (TOP-REAL n) st S1[x,y] proof let x be Element of T; ::_thesis: ex y being Element of (TOP-REAL n) st S1[x,y] f . x in REAL by XREAL_0:def_1; then n |-> (f . x) is Element of REAL n by FINSEQ_2:112; then reconsider y = n |-> (f . x) as Point of (TOP-REAL n) by EUCLID:22; take y ; ::_thesis: S1[x,y] thus S1[x,y] ; ::_thesis: verum end; ex F being Function of T,(TOP-REAL n) st for x being Element of T holds S1[x,F . x] from FUNCT_2:sch_3(A1); hence ex b1 being Function of T,(TOP-REAL n) st for t being Element of T holds b1 . t = n |-> (f . t) ; ::_thesis: verum end; uniqueness for b1, b2 being Function of T,(TOP-REAL n) st ( for t being Element of T holds b1 . t = n |-> (f . t) ) & ( for t being Element of T holds b2 . t = n |-> (f . t) ) holds b1 = b2 proof let F, G be Function of T,(TOP-REAL n); ::_thesis: ( ( for t being Element of T holds F . t = n |-> (f . t) ) & ( for t being Element of T holds G . t = n |-> (f . t) ) implies F = G ) assume that A2: for t being Element of T holds F . t = n |-> (f . t) and A3: for t being Element of T holds G . t = n |-> (f . t) ; ::_thesis: F = G let t be Element of T; :: according to FUNCT_2:def_8 ::_thesis: F . t = G . t thus F . t = n |-> (f . t) by A2 .= G . t by A3 ; ::_thesis: verum end; end; :: deftheorem Def4 defines incl TOPREALC:def_4_:_ for n being Nat for T being non empty set for R being real-membered set for f being Function of T,R for b5 being Function of T,(TOP-REAL n) holds ( b5 = incl (f,n) iff for t being Element of T holds b5 . t = n |-> (f . t) ); theorem Th47: :: TOPREALC:47 for x being set for n being Nat for T being non empty TopSpace for R being real-membered set for f being Function of T,R for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t proof let x be set ; ::_thesis: for n being Nat for T being non empty TopSpace for R being real-membered set for f being Function of T,R for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t let n be Nat; ::_thesis: for T being non empty TopSpace for R being real-membered set for f being Function of T,R for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t let T be non empty TopSpace; ::_thesis: for R being real-membered set for f being Function of T,R for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t let R be real-membered set ; ::_thesis: for f being Function of T,R for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t let f be Function of T,R; ::_thesis: for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t let t be Point of T; ::_thesis: ( x in Seg n implies ((incl (f,n)) . t) . x = f . t ) assume A1: x in Seg n ; ::_thesis: ((incl (f,n)) . t) . x = f . t thus ((incl (f,n)) . t) . x = (n |-> (f . t)) . x by Def4 .= f . t by A1, FINSEQ_2:57 ; ::_thesis: verum end; theorem Th48: :: TOPREALC:48 for T being non empty set for R being real-membered set for f being Function of T,R holds incl (f,0) = T --> 0 proof let T be non empty set ; ::_thesis: for R being real-membered set for f being Function of T,R holds incl (f,0) = T --> 0 let R be real-membered set ; ::_thesis: for f being Function of T,R holds incl (f,0) = T --> 0 let f be Function of T,R; ::_thesis: incl (f,0) = T --> 0 reconsider z = 0 as Element of (TOP-REAL 0) ; incl (f,0) = T --> z proof let x be Element of T; :: according to FUNCT_2:def_8 ::_thesis: (incl (f,0)) . x = (T --> z) . x thus (incl (f,0)) . x = (T --> z) . x ; ::_thesis: verum end; hence incl (f,0) = T --> 0 ; ::_thesis: verum end; theorem Th49: :: TOPREALC:49 for n being Nat for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <+> g = f <++> (incl (g,n)) proof let n be Nat; ::_thesis: for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <+> g = f <++> (incl (g,n)) let T be non empty TopSpace; ::_thesis: for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <+> g = f <++> (incl (g,n)) let f be Function of T,(TOP-REAL n); ::_thesis: for g being Function of T,R^1 holds f <+> g = f <++> (incl (g,n)) let g be Function of T,R^1; ::_thesis: f <+> g = f <++> (incl (g,n)) set I = incl (g,n); reconsider h = f <+> g as Function of T,(TOP-REAL n) by Th43; reconsider G = f <++> (incl (g,n)) as Function of T,(TOP-REAL n) by Th39; h = G proof let t be Point of T; :: according to FUNCT_2:def_8 ::_thesis: h . t = G . t A1: dom h = the carrier of T by FUNCT_2:def_1; A2: (f . t) + ((incl (g,n)) . t) = (f . t) + (g . t) proof A3: ( dom (f . t) = Seg n & dom ((incl (g,n)) . t) = Seg n ) by FINSEQ_1:89; A4: dom ((f . t) + ((incl (g,n)) . t)) = (dom (f . t)) /\ (dom ((incl (g,n)) . t)) by VALUED_1:def_1 .= Seg n by A3 ; A5: dom ((f . t) + (g . t)) = dom (f . t) by VALUED_1:def_2; thus dom ((f . t) + ((incl (g,n)) . t)) = dom ((f . t) + (g . t)) by A4, FINSEQ_1:89; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f . t) + ((incl (g,n)) . t)) or ((f . t) + ((incl (g,n)) . t)) . b1 = ((f . t) + (g . t)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f . t) + ((incl (g,n)) . t)) or ((f . t) + ((incl (g,n)) . t)) . x = ((f . t) + (g . t)) . x ) assume A6: x in dom ((f . t) + ((incl (g,n)) . t)) ; ::_thesis: ((f . t) + ((incl (g,n)) . t)) . x = ((f . t) + (g . t)) . x hence ((f . t) + ((incl (g,n)) . t)) . x = ((f . t) . x) + (((incl (g,n)) . t) . x) by VALUED_1:def_1 .= ((f . t) . x) + (g . t) by A4, A6, Th47 .= ((f . t) + (g . t)) . x by A3, A4, A6, A5, VALUED_1:def_2 ; ::_thesis: verum end; dom G = the carrier of T by FUNCT_2:def_1; hence G . t = (f . t) + ((incl (g,n)) . t) by VALUED_2:def_45 .= h . t by A1, A2, VALUED_2:def_41 ; ::_thesis: verum end; hence f <+> g = f <++> (incl (g,n)) ; ::_thesis: verum end; theorem Th50: :: TOPREALC:50 for n being Nat for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <-> g = f <--> (incl (g,n)) proof let n be Nat; ::_thesis: for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <-> g = f <--> (incl (g,n)) let T be non empty TopSpace; ::_thesis: for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <-> g = f <--> (incl (g,n)) let f be Function of T,(TOP-REAL n); ::_thesis: for g being Function of T,R^1 holds f <-> g = f <--> (incl (g,n)) let g be Function of T,R^1; ::_thesis: f <-> g = f <--> (incl (g,n)) set I = incl (g,n); reconsider h = f <-> g as Function of T,(TOP-REAL n) by Th44; reconsider G = f <--> (incl (g,n)) as Function of T,(TOP-REAL n) by Th40; h = G proof let t be Point of T; :: according to FUNCT_2:def_8 ::_thesis: h . t = G . t A1: dom h = the carrier of T by FUNCT_2:def_1; A2: (f . t) - ((incl (g,n)) . t) = (f . t) - (g . t) proof A3: ( dom (f . t) = Seg n & dom ((incl (g,n)) . t) = Seg n ) by FINSEQ_1:89; A4: dom ((f . t) - ((incl (g,n)) . t)) = (dom (f . t)) /\ (dom ((incl (g,n)) . t)) by VALUED_1:12 .= Seg n by A3 ; A5: dom ((f . t) - (g . t)) = dom (f . t) by VALUED_1:def_2; thus dom ((f . t) - ((incl (g,n)) . t)) = dom ((f . t) - (g . t)) by A4, FINSEQ_1:89; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f . t) - ((incl (g,n)) . t)) or ((f . t) - ((incl (g,n)) . t)) . b1 = ((f . t) - (g . t)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f . t) - ((incl (g,n)) . t)) or ((f . t) - ((incl (g,n)) . t)) . x = ((f . t) - (g . t)) . x ) assume A6: x in dom ((f . t) - ((incl (g,n)) . t)) ; ::_thesis: ((f . t) - ((incl (g,n)) . t)) . x = ((f . t) - (g . t)) . x hence ((f . t) - ((incl (g,n)) . t)) . x = ((f . t) . x) - (((incl (g,n)) . t) . x) by VALUED_1:13 .= ((f . t) . x) - (g . t) by A4, A6, Th47 .= ((f . t) - (g . t)) . x by A3, A4, A6, A5, VALUED_1:def_2 ; ::_thesis: verum end; dom G = the carrier of T by FUNCT_2:def_1; hence G . t = (f . t) - ((incl (g,n)) . t) by VALUED_2:def_46 .= h . t by A1, A2, VALUED_2:62 ; ::_thesis: verum end; hence f <-> g = f <--> (incl (g,n)) ; ::_thesis: verum end; theorem Th51: :: TOPREALC:51 for n being Nat for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <#> g = f <##> (incl (g,n)) proof let n be Nat; ::_thesis: for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <#> g = f <##> (incl (g,n)) let T be non empty TopSpace; ::_thesis: for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <#> g = f <##> (incl (g,n)) let f be Function of T,(TOP-REAL n); ::_thesis: for g being Function of T,R^1 holds f <#> g = f <##> (incl (g,n)) let g be Function of T,R^1; ::_thesis: f <#> g = f <##> (incl (g,n)) set I = incl (g,n); reconsider h = f <#> g as Function of T,(TOP-REAL n) by Th45; reconsider G = f <##> (incl (g,n)) as Function of T,(TOP-REAL n) by Th41; h = G proof let t be Point of T; :: according to FUNCT_2:def_8 ::_thesis: h . t = G . t A1: dom h = the carrier of T by FUNCT_2:def_1; A2: (f . t) (#) ((incl (g,n)) . t) = (f . t) (#) (g . t) proof A3: ( dom (f . t) = Seg n & dom ((incl (g,n)) . t) = Seg n ) by FINSEQ_1:89; A4: dom ((f . t) (#) ((incl (g,n)) . t)) = (dom (f . t)) /\ (dom ((incl (g,n)) . t)) by VALUED_1:def_4 .= Seg n by A3 ; hence dom ((f . t) (#) ((incl (g,n)) . t)) = dom ((f . t) (#) (g . t)) by FINSEQ_1:89; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f . t) (#) ((incl (g,n)) . t)) or ((f . t) (#) ((incl (g,n)) . t)) . b1 = ((f . t) (#) (g . t)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f . t) (#) ((incl (g,n)) . t)) or ((f . t) (#) ((incl (g,n)) . t)) . x = ((f . t) (#) (g . t)) . x ) assume A5: x in dom ((f . t) (#) ((incl (g,n)) . t)) ; ::_thesis: ((f . t) (#) ((incl (g,n)) . t)) . x = ((f . t) (#) (g . t)) . x hence ((f . t) (#) ((incl (g,n)) . t)) . x = ((f . t) . x) * (((incl (g,n)) . t) . x) by VALUED_1:def_4 .= ((f . t) . x) * (g . t) by A4, A5, Th47 .= ((f . t) (#) (g . t)) . x by VALUED_1:6 ; ::_thesis: verum end; dom G = the carrier of T by FUNCT_2:def_1; hence G . t = (f . t) (#) ((incl (g,n)) . t) by VALUED_2:def_47 .= h . t by A1, A2, VALUED_2:def_43 ; ::_thesis: verum end; hence f <#> g = f <##> (incl (g,n)) ; ::_thesis: verum end; theorem :: TOPREALC:52 for n being Nat for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f g = f (incl (g,n)) proof let n be Nat; ::_thesis: for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f g = f (incl (g,n)) let T be non empty TopSpace; ::_thesis: for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f g = f (incl (g,n)) let f be Function of T,(TOP-REAL n); ::_thesis: for g being Function of T,R^1 holds f g = f (incl (g,n)) let g be Function of T,R^1; ::_thesis: f g = f (incl (g,n)) set I = incl (g,n); reconsider h = f g as Function of T,(TOP-REAL n) by Th46; reconsider G = f (incl (g,n)) as Function of T,(TOP-REAL n) by Th42; h = G proof let t be Point of T; :: according to FUNCT_2:def_8 ::_thesis: h . t = G . t A1: dom h = the carrier of T by FUNCT_2:def_1; A2: (f . t) /" ((incl (g,n)) . t) = (f . t) (#) ((g ") . t) proof A3: ( dom (f . t) = Seg n & dom ((incl (g,n)) . t) = Seg n ) by FINSEQ_1:89; A4: dom ((f . t) /" ((incl (g,n)) . t)) = (dom (f . t)) /\ (dom ((incl (g,n)) . t)) by VALUED_1:16 .= Seg n by A3 ; hence dom ((f . t) /" ((incl (g,n)) . t)) = dom ((f . t) (#) ((g ") . t)) by FINSEQ_1:89; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f . t) /" ((incl (g,n)) . t)) or ((f . t) /" ((incl (g,n)) . t)) . b1 = ((f . t) (#) ((g ") . t)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f . t) /" ((incl (g,n)) . t)) or ((f . t) /" ((incl (g,n)) . t)) . x = ((f . t) (#) ((g ") . t)) . x ) assume A5: x in dom ((f . t) /" ((incl (g,n)) . t)) ; ::_thesis: ((f . t) /" ((incl (g,n)) . t)) . x = ((f . t) (#) ((g ") . t)) . x thus ((f . t) /" ((incl (g,n)) . t)) . x = ((f . t) . x) / (((incl (g,n)) . t) . x) by VALUED_1:17 .= ((f . t) . x) / (g . t) by A4, A5, Th47 .= ((f . t) . x) * ((g ") . t) by VALUED_1:10 .= ((f . t) (#) ((g ") . t)) . x by VALUED_1:6 ; ::_thesis: verum end; dom G = the carrier of T by FUNCT_2:def_1; hence G . t = (f . t) /" ((incl (g,n)) . t) by VALUED_2:def_48 .= h . t by A1, A2, VALUED_2:def_43 ; ::_thesis: verum end; hence f g = f (incl (g,n)) ; ::_thesis: verum end; definition let n be Nat; set T = TOP-REAL n; set c = the carrier of (TOP-REAL n); A1: the carrier of [:(TOP-REAL n),(TOP-REAL n):] = [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] by BORSUK_1:def_2; func TIMES n -> Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n) means :Def5: :: TOPREALC:def 5 for x, y being Point of (TOP-REAL n) holds it . (x,y) = x (#) y; existence ex b1 being Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n) st for x, y being Point of (TOP-REAL n) holds b1 . (x,y) = x (#) y proof deffunc H1( Point of (TOP-REAL n), Point of (TOP-REAL n)) -> Point of (TOP-REAL n) = \$1 (#) \$2; ex f being Function of [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n) st for x, y being Element of the carrier of (TOP-REAL n) holds f . (x,y) = H1(x,y) from BINOP_1:sch_4(); hence ex b1 being Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n) st for x, y being Point of (TOP-REAL n) holds b1 . (x,y) = x (#) y by A1; ::_thesis: verum end; uniqueness for b1, b2 being Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n) st ( for x, y being Point of (TOP-REAL n) holds b1 . (x,y) = x (#) y ) & ( for x, y being Point of (TOP-REAL n) holds b2 . (x,y) = x (#) y ) holds b1 = b2 proof let f, g be Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n); ::_thesis: ( ( for x, y being Point of (TOP-REAL n) holds f . (x,y) = x (#) y ) & ( for x, y being Point of (TOP-REAL n) holds g . (x,y) = x (#) y ) implies f = g ) assume that A2: for x, y being Point of (TOP-REAL n) holds f . (x,y) = x (#) y and A3: for x, y being Point of (TOP-REAL n) holds g . (x,y) = x (#) y ; ::_thesis: f = g now__::_thesis:_for_x,_y_being_Point_of_(TOP-REAL_n)_holds_f_._(x,y)_=_g_._(x,y) let x, y be Point of (TOP-REAL n); ::_thesis: f . (x,y) = g . (x,y) thus f . (x,y) = x (#) y by A2 .= g . (x,y) by A3 ; ::_thesis: verum end; hence f = g by A1, BINOP_1:2; ::_thesis: verum end; end; :: deftheorem Def5 defines TIMES TOPREALC:def_5_:_ for n being Nat for b2 being Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n) holds ( b2 = TIMES n iff for x, y being Point of (TOP-REAL n) holds b2 . (x,y) = x (#) y ); theorem Th53: :: TOPREALC:53 TIMES 0 = [:(TOP-REAL 0),(TOP-REAL 0):] --> (0. (TOP-REAL 0)) proof set T = TOP-REAL 0; let x be Element of the carrier of [:(TOP-REAL 0),(TOP-REAL 0):]; :: according to FUNCT_2:def_8 ::_thesis: (TIMES 0) . x = ([:(TOP-REAL 0),(TOP-REAL 0):] --> (0. (TOP-REAL 0))) . x thus (TIMES 0) . x = ([:(TOP-REAL 0),(TOP-REAL 0):] --> (0. (TOP-REAL 0))) . x ; ::_thesis: verum end; theorem Th54: :: TOPREALC:54 for n being Nat for T being non empty TopSpace for f, g being Function of T,(TOP-REAL n) holds f <##> g = (TIMES n) .: (f,g) proof let n be Nat; ::_thesis: for T being non empty TopSpace for f, g being Function of T,(TOP-REAL n) holds f <##> g = (TIMES n) .: (f,g) let T be non empty TopSpace; ::_thesis: for f, g being Function of T,(TOP-REAL n) holds f <##> g = (TIMES n) .: (f,g) set R = TOP-REAL n; set F = TIMES n; let f, g be Function of T,(TOP-REAL n); ::_thesis: f <##> g = (TIMES n) .: (f,g) A1: dom (f <##> g) = (dom f) /\ (dom g) by VALUED_2:def_47; dom (TIMES n) = the carrier of [:(TOP-REAL n),(TOP-REAL n):] by FUNCT_2:def_1 .= [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] by BORSUK_1:def_2 ; then [:(rng f),(rng g):] c= dom (TIMES n) by ZFMISC_1:96; then A2: dom ((TIMES n) .: (f,g)) = (dom f) /\ (dom g) by FUNCOP_1:69; now__::_thesis:_for_x_being_set_st_x_in_dom_(f_<##>_g)_holds_ (f_<##>_g)_._x_=_((TIMES_n)_.:_(f,g))_._x let x be set ; ::_thesis: ( x in dom (f <##> g) implies (f <##> g) . x = ((TIMES n) .: (f,g)) . x ) assume A3: x in dom (f <##> g) ; ::_thesis: (f <##> g) . x = ((TIMES n) .: (f,g)) . x then A4: ( f . x is Point of (TOP-REAL n) & g . x is Point of (TOP-REAL n) ) by FUNCT_2:5; thus (f <##> g) . x = (f . x) (#) (g . x) by A3, VALUED_2:def_47 .= (TIMES n) . ((f . x),(g . x)) by A4, Def5 .= ((TIMES n) .: (f,g)) . x by A1, A2, A3, FUNCOP_1:22 ; ::_thesis: verum end; hence f <##> g = (TIMES n) .: (f,g) by A1, A2, FUNCT_1:2; ::_thesis: verum end; definition let m, n be Nat; A1: ( m in NAT & n in NAT ) by ORDINAL1:def_12; func PROJ (m,n) -> Function of (TOP-REAL m),R^1 means :Def6: :: TOPREALC:def 6 for p being Element of (TOP-REAL m) holds it . p = p /. n; existence ex b1 being Function of (TOP-REAL m),R^1 st for p being Element of (TOP-REAL m) holds b1 . p = p /. n by A1, JORDAN2B:1; uniqueness for b1, b2 being Function of (TOP-REAL m),R^1 st ( for p being Element of (TOP-REAL m) holds b1 . p = p /. n ) & ( for p being Element of (TOP-REAL m) holds b2 . p = p /. n ) holds b1 = b2 proof let F, G be Function of (TOP-REAL m),R^1; ::_thesis: ( ( for p being Element of (TOP-REAL m) holds F . p = p /. n ) & ( for p being Element of (TOP-REAL m) holds G . p = p /. n ) implies F = G ) assume that A2: for p being Element of (TOP-REAL m) holds F . p = p /. n and A3: for p being Element of (TOP-REAL m) holds G . p = p /. n ; ::_thesis: F = G let p be Element of (TOP-REAL m); :: according to FUNCT_2:def_8 ::_thesis: F . p = G . p thus F . p = p /. n by A2 .= G . p by A3 ; ::_thesis: verum end; end; :: deftheorem Def6 defines PROJ TOPREALC:def_6_:_ for m, n being Nat for b3 being Function of (TOP-REAL m),R^1 holds ( b3 = PROJ (m,n) iff for p being Element of (TOP-REAL m) holds b3 . p = p /. n ); theorem Th55: :: TOPREALC:55 for m, n being Nat for r being real number for p being Point of (TOP-REAL m) st n in dom p holds (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ proof let m, n be Nat; ::_thesis: for r being real number for p being Point of (TOP-REAL m) st n in dom p holds (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ let r be real number ; ::_thesis: for p being Point of (TOP-REAL m) st n in dom p holds (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ let p be Point of (TOP-REAL m); ::_thesis: ( n in dom p implies (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ ) assume A1: n in dom p ; ::_thesis: (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ A2: m in NAT by ORDINAL1:def_12; percases ( r <= 0 or 0 < r ) ; supposeA3: r <= 0 ; ::_thesis: (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ then ].((p /. n) - r),((p /. n) + r).[ = {} ; hence (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ by A3; ::_thesis: verum end; supposeA4: 0 < r ; ::_thesis: (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ A5: dom p = Seg m by FINSEQ_1:89; thus (PROJ (m,n)) .: (Ball (p,r)) c= ].((p /. n) - r),((p /. n) + r).[ :: according to XBOOLE_0:def_10 ::_thesis: ].((p /. n) - r),((p /. n) + r).[ c= (PROJ (m,n)) .: (Ball (p,r)) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (PROJ (m,n)) .: (Ball (p,r)) or y in ].((p /. n) - r),((p /. n) + r).[ ) assume y in (PROJ (m,n)) .: (Ball (p,r)) ; ::_thesis: y in ].((p /. n) - r),((p /. n) + r).[ then consider x being Element of (TOP-REAL m) such that A6: x in Ball (p,r) and A7: y = (PROJ (m,n)) . x by FUNCT_2:65; A8: (PROJ (m,n)) . x = x /. n by Def6; A9: |.(x - p).| < r by A2, A6, TOPREAL9:7; 0 <= Sum (sqr (x - p)) by RVSUM_1:86; then (sqrt (Sum (sqr (x - p)))) ^2 = Sum (sqr (x - p)) by SQUARE_1:def_2; then A10: Sum (sqr (x - p)) < r ^2 by A9, SQUARE_1:16; dom x = Seg m by FINSEQ_1:89; then ((x /. n) - (p /. n)) ^2 <= Sum (sqr (x - p)) by A5, A1, EUCLID_9:8; then ((x /. n) - (p /. n)) ^2 < r ^2 by A10, XXREAL_0:2; then ( - r < (x /. n) - (p /. n) & (x /. n) - (p /. n) < r ) by A4, SQUARE_1:48; then ( (- r) + (p /. n) < ((x /. n) - (p /. n)) + (p /. n) & ((x /. n) - (p /. n)) + (p /. n) < r + (p /. n) ) by XREAL_1:6; hence y in ].((p /. n) - r),((p /. n) + r).[ by A7, A8, XXREAL_1:4; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in ].((p /. n) - r),((p /. n) + r).[ or y in (PROJ (m,n)) .: (Ball (p,r)) ) assume A11: y in ].((p /. n) - r),((p /. n) + r).[ ; ::_thesis: y in (PROJ (m,n)) .: (Ball (p,r)) then reconsider y = y as Element of REAL ; set x = p +* (n,y); reconsider X = p +* (n,y) as FinSequence of REAL by EUCLID:24; A12: dom X = dom p by FUNCT_7:30; A13: p /. n = p . n by A1, PARTFUN1:def_6; ( (p /. n) - r < y & y < (p /. n) + r ) by A11, XXREAL_1:4; then A14: ( y - (p /. n) < r & - r < y - (p /. n) ) by XREAL_1:19, XREAL_1:20; (p +* (n,y)) - p = (0* m) +* (n,(y - (p . n))) by Th17; then |.((p +* (n,y)) - p).| = abs (y - (p . n)) by A1, A5, Th13; then |.((p +* (n,y)) - p).| < r by A13, A14, SEQ_2:1; then A15: p +* (n,y) in Ball (p,r) by A2, TOPREAL9:7; (PROJ (m,n)) . (p +* (n,y)) = (p +* (n,y)) /. n by Def6 .= X . n by A12, A1, PARTFUN1:def_6 .= y by A1, FUNCT_7:31 ; hence y in (PROJ (m,n)) .: (Ball (p,r)) by A15, FUNCT_2:35; ::_thesis: verum end; end; end; theorem :: TOPREALC:56 for T being non empty TopSpace for m being non zero Nat for f being Function of T,R^1 holds f = (PROJ (m,m)) * (incl (f,m)) proof let T be non empty TopSpace; ::_thesis: for m being non zero Nat for f being Function of T,R^1 holds f = (PROJ (m,m)) * (incl (f,m)) let m be non zero Nat; ::_thesis: for f being Function of T,R^1 holds f = (PROJ (m,m)) * (incl (f,m)) let f be Function of T,R^1; ::_thesis: f = (PROJ (m,m)) * (incl (f,m)) let p be Point of T; :: according to FUNCT_2:def_8 ::_thesis: f . p = ((PROJ (m,m)) * (incl (f,m))) . p set I = incl (f,m); reconsider G = m |-> (f . p) as FinSequence of REAL ; A1: m in NAT by ORDINAL1:def_12; 1 <= m by NAT_1:14; then A2: m in Seg m by A1; A3: dom (m |-> (f . p)) = Seg m by FUNCOP_1:13; thus ((PROJ (m,m)) * (incl (f,m))) . p = (PROJ (m,m)) . ((incl (f,m)) . p) by FUNCT_2:15 .= ((incl (f,m)) . p) /. m by Def6 .= G /. m by Def4 .= G . m by A2, A3, PARTFUN1:def_6 .= f . p by A2, FINSEQ_2:57 ; ::_thesis: verum end; begin registration let T be non empty TopSpace; cluster Relation-like non-empty the carrier of T -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of T, the carrier of R^1) complex-valued ext-real-valued real-valued continuous for Element of bool [: the carrier of T, the carrier of R^1:]; existence ex b1 being Function of T,R^1 st ( b1 is non-empty & b1 is continuous ) proof take T --> (R^1 1) ; ::_thesis: ( T --> (R^1 1) is non-empty & T --> (R^1 1) is continuous ) thus ( T --> (R^1 1) is non-empty & T --> (R^1 1) is continuous ) ; ::_thesis: verum end; end; theorem :: TOPREALC:57 for n, m being Nat st n in Seg m holds PROJ (m,n) is continuous proof let n, m be Nat; ::_thesis: ( n in Seg m implies PROJ (m,n) is continuous ) assume A1: n in Seg m ; ::_thesis: PROJ (m,n) is continuous A2: m in NAT by ORDINAL1:def_12; for p being Element of (TOP-REAL m) holds (PROJ (m,n)) . p = p /. n by Def6; hence PROJ (m,n) is continuous by A2, A1, JORDAN2B:18; ::_thesis: verum end; theorem :: TOPREALC:58 for n, m being Nat st n in Seg m holds PROJ (m,n) is open proof let n, m be Nat; ::_thesis: ( n in Seg m implies PROJ (m,n) is open ) set f = PROJ (m,n); assume A1: n in Seg m ; ::_thesis: PROJ (m,n) is open for p being Point of (TOP-REAL m) for r being positive real number ex s being positive real number st ].(((PROJ (m,n)) . p) - s),(((PROJ (m,n)) . p) + s).[ c= (PROJ (m,n)) .: (Ball (p,r)) proof let p be Point of (TOP-REAL m); ::_thesis: for r being positive real number ex s being positive real number st ].(((PROJ (m,n)) . p) - s),(((PROJ (m,n)) . p) + s).[ c= (PROJ (m,n)) .: (Ball (p,r)) let r be positive real number ; ::_thesis: ex s being positive real number st ].(((PROJ (m,n)) . p) - s),(((PROJ (m,n)) . p) + s).[ c= (PROJ (m,n)) .: (Ball (p,r)) take r ; ::_thesis: ].(((PROJ (m,n)) . p) - r),(((PROJ (m,n)) . p) + r).[ c= (PROJ (m,n)) .: (Ball (p,r)) A2: dom p = Seg m by FINSEQ_1:89; p /. n = (PROJ (m,n)) . p by Def6; hence ].(((PROJ (m,n)) . p) - r),(((PROJ (m,n)) . p) + r).[ c= (PROJ (m,n)) .: (Ball (p,r)) by A2, A1, Th55; ::_thesis: verum end; hence PROJ (m,n) is open by TOPS_4:13; ::_thesis: verum end; registration let n be Nat; let T be non empty TopSpace; let f be continuous Function of T,R^1; cluster incl (f,n) -> continuous ; coherence incl (f,n) is continuous proof reconsider n = n as Element of NAT by ORDINAL1:def_12; set g = incl (f,n); percases ( n = 0 or n > 0 ) ; supposeA1: n = 0 ; ::_thesis: incl (f,n) is continuous then reconsider z = 0 as Element of (TOP-REAL n) ; incl (f,0) = T --> z by Th48; hence incl (f,n) is continuous by A1; ::_thesis: verum end; supposeA2: n > 0 ; ::_thesis: incl (f,n) is continuous for p being Point of T for V being Subset of (TOP-REAL n) st (incl (f,n)) . p in V & V is open holds ex W being Subset of T st ( p in W & W is open & (incl (f,n)) .: W c= V ) proof let p be Point of T; ::_thesis: for V being Subset of (TOP-REAL n) st (incl (f,n)) . p in V & V is open holds ex W being Subset of T st ( p in W & W is open & (incl (f,n)) .: W c= V ) let V be Subset of (TOP-REAL n); ::_thesis: ( (incl (f,n)) . p in V & V is open implies ex W being Subset of T st ( p in W & W is open & (incl (f,n)) .: W c= V ) ) assume that A3: (incl (f,n)) . p in V and A4: V is open ; ::_thesis: ex W being Subset of T st ( p in W & W is open & (incl (f,n)) .: W c= V ) A5: (incl (f,n)) . p in Int V by A3, A4, TOPS_1:23; reconsider s = (incl (f,n)) . p as Point of (Euclid n) by EUCLID:67; consider r being real number such that A6: r > 0 and A7: Ball (s,r) c= V by A5, GOBOARD6:5; reconsider G = n |-> (f . p) as FinSequence of REAL ; 1 <= n by A2, NAT_1:14; then A8: n in Seg n ; reconsider F = (incl (f,n)) . p as FinSequence of REAL by EUCLID:24; A9: F = n |-> (f . p) by Def4; A10: dom (n |-> (f . p)) = Seg n by FUNCOP_1:13; A11: ((incl (f,n)) . p) /. n = G /. n by Def4 .= G . n by A8, A10, PARTFUN1:def_6 .= f . p by A8, FINSEQ_2:57 ; set R = r / (sqrt n); set RR = R^1 ].((((incl (f,n)) . p) /. n) - (r / (sqrt n))),((((incl (f,n)) . p) /. n) + (r / (sqrt n))).[; f . p in R^1 ].((((incl (f,n)) . p) /. n) - (r / (sqrt n))),((((incl (f,n)) . p) /. n) + (r / (sqrt n))).[ by A2, A11, A6, TOPREAL6:15; then consider W being Subset of T such that A12: ( p in W & W is open ) and A13: f .: W c= R^1 ].((((incl (f,n)) . p) /. n) - (r / (sqrt n))),((((incl (f,n)) . p) /. n) + (r / (sqrt n))).[ by JGRAPH_2:10; take W ; ::_thesis: ( p in W & W is open & (incl (f,n)) .: W c= V ) thus ( p in W & W is open ) by A12; ::_thesis: (incl (f,n)) .: W c= V let y be Element of (TOP-REAL n); :: according to LATTICE7:def_1 ::_thesis: ( not y in (incl (f,n)) .: W or y in V ) assume y in (incl (f,n)) .: W ; ::_thesis: y in V then consider x being Element of T such that A14: x in W and A15: (incl (f,n)) . x = y by FUNCT_2:65; reconsider y1 = y as Point of (Euclid n) by EUCLID:67; reconsider y2 = y1, s2 = s as Element of REAL n ; A16: y2 = n |-> (f . x) by A15, Def4; A17: (Pitag_dist n) . (y1,s) = |.(y2 - s2).| by EUCLID:def_6 .= (sqrt n) * (abs ((f . x) - (f . p))) by A16, A9, Th11 ; f . x in f .: W by A14, FUNCT_2:35; then abs ((f . x) - (f . p)) < r / (sqrt n) by A13, A11, RCOMP_1:1; then dist (y1,s) < r by A2, A17, XREAL_1:79; then y in Ball (s,r) by METRIC_1:11; hence y in V by A7; ::_thesis: verum end; hence incl (f,n) is continuous by JGRAPH_2:10; ::_thesis: verum end; end; end; end; registration let n be Nat; cluster TIMES n -> continuous ; coherence TIMES n is continuous proof percases ( not n is zero or n is zero ) ; suppose not n is zero ; ::_thesis: TIMES n is continuous then reconsider n = n as non zero Element of NAT by ORDINAL1:def_12; set T = TOP-REAL n; set F = TIMES n; set J = (Seg n) --> R^1; set c = the carrier of (TOP-REAL n); A1: TopSpaceMetr (Euclid n) = product ((Seg n) --> R^1) by EUCLID_9:28; A2: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8; then reconsider F = TIMES n as Function of [:(TOP-REAL n),(TOP-REAL n):],(product ((Seg n) --> R^1)) by EUCLID_9:28; A3: the carrier of (TOP-REAL n) = REAL n by EUCLID:22; now__::_thesis:_for_i_being_Element_of_Seg_n_holds_(proj_(((Seg_n)_-->_R^1),i))_*_F_is_continuous let i be Element of Seg n; ::_thesis: (proj (((Seg n) --> R^1),i)) * F is continuous set P = proj (((Seg n) --> R^1),i); A4: ((Seg n) --> R^1) . i = R^1 by FUNCOP_1:7; reconsider f = (proj (((Seg n) --> R^1),i)) * F as Function of [:(TOP-REAL n),(TOP-REAL n):],R^1 by FUNCOP_1:7; A5: the carrier of [:(TOP-REAL n),(TOP-REAL n):] = [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] by BORSUK_1:def_2; A6: for a, b being Point of (TOP-REAL n) holds f . (a,b) = (a . i) * (b . i) proof let a, b be Point of (TOP-REAL n); ::_thesis: f . (a,b) = (a . i) * (b . i) thus f . (a,b) = (proj (((Seg n) --> R^1),i)) . (F . (a,b)) by A5, BINOP_1:18 .= (proj (((Seg n) --> R^1),i)) . (a (#) b) by Def5 .= (a (#) b) . i by A1, A2, YELLOW17:8 .= (a . i) * (b . i) by VALUED_1:5 ; ::_thesis: verum end; deffunc H1( Element of the carrier of (TOP-REAL n), Element of the carrier of (TOP-REAL n)) -> Element of REAL = n . i; consider f1 being Function of [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):],REAL such that A7: for a, b being Element of the carrier of (TOP-REAL n) holds f1 . (a,b) = H1(a,b) from BINOP_1:sch_4(); reconsider f1 = f1 as Function of [:(TOP-REAL n),(TOP-REAL n):],R^1 by A5; deffunc H2( Element of the carrier of (TOP-REAL n), Element of the carrier of (TOP-REAL n)) -> Element of REAL = c2 . i; consider f2 being Function of [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):],REAL such that A8: for a, b being Element of the carrier of (TOP-REAL n) holds f2 . (a,b) = H2(a,b) from BINOP_1:sch_4(); reconsider f1 = f1, f2 = f2 as Function of [:(TOP-REAL n),(TOP-REAL n):],R^1 by A5; for p being Point of [:(TOP-REAL n),(TOP-REAL n):] for r being positive real number ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f1 .: W c= ].((f1 . p) - r),((f1 . p) + r).[ ) proof let p be Point of [:(TOP-REAL n),(TOP-REAL n):]; ::_thesis: for r being positive real number ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f1 .: W c= ].((f1 . p) - r),((f1 . p) + r).[ ) let r be positive real number ; ::_thesis: ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f1 .: W c= ].((f1 . p) - r),((f1 . p) + r).[ ) consider p1, p2 being Point of (TOP-REAL n) such that A9: p = [p1,p2] by BORSUK_1:10; set B1 = Ball (p1,r); set B2 = [#] (TOP-REAL n); reconsider W = [:(Ball (p1,r)),([#] (TOP-REAL n)):] as open Subset of [:(TOP-REAL n),(TOP-REAL n):] by BORSUK_1:6; take W ; ::_thesis: ( p in W & f1 .: W c= ].((f1 . p) - r),((f1 . p) + r).[ ) ( p1 in Ball (p1,r) & p2 in [#] (TOP-REAL n) ) by TOPGEN_5:13; hence p in W by A9, ZFMISC_1:def_2; ::_thesis: f1 .: W c= ].((f1 . p) - r),((f1 . p) + r).[ let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f1 .: W or y in ].((f1 . p) - r),((f1 . p) + r).[ ) assume y in f1 .: W ; ::_thesis: y in ].((f1 . p) - r),((f1 . p) + r).[ then consider x being Point of [:(TOP-REAL n),(TOP-REAL n):] such that A10: x in W and A11: f1 . x = y by FUNCT_2:65; consider x1, x2 being Point of (TOP-REAL n) such that A12: x = [x1,x2] by BORSUK_1:10; A13: ( f1 . (x1,x2) = x1 . i & f1 . (p1,p2) = p1 . i ) by A7; x1 in Ball (p1,r) by A10, A12, ZFMISC_1:87; then A14: |.(x1 - p1).| < r by TOPREAL9:7; dom (x1 - p1) = Seg n by FINSEQ_1:89; then (x1 . i) - (p1 . i) = (x1 - p1) . i by VALUED_1:13; then abs ((x1 . i) - (p1 . i)) <= |.(x1 - p1).| by A3, REAL_NS1:8; then abs ((f1 . x) - (f1 . p)) < r by A9, A12, A13, A14, XXREAL_0:2; hence y in ].((f1 . p) - r),((f1 . p) + r).[ by A11, RCOMP_1:1; ::_thesis: verum end; then A15: f1 is continuous by TOPS_4:21; for p being Point of [:(TOP-REAL n),(TOP-REAL n):] for r being positive real number ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f2 .: W c= ].((f2 . p) - r),((f2 . p) + r).[ ) proof let p be Point of [:(TOP-REAL n),(TOP-REAL n):]; ::_thesis: for r being positive real number ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f2 .: W c= ].((f2 . p) - r),((f2 . p) + r).[ ) let r be positive real number ; ::_thesis: ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f2 .: W c= ].((f2 . p) - r),((f2 . p) + r).[ ) consider p1, p2 being Point of (TOP-REAL n) such that A16: p = [p1,p2] by BORSUK_1:10; set B1 = [#] (TOP-REAL n); set B2 = Ball (p2,r); reconsider W = [:([#] (TOP-REAL n)),(Ball (p2,r)):] as open Subset of [:(TOP-REAL n),(TOP-REAL n):] by BORSUK_1:6; take W ; ::_thesis: ( p in W & f2 .: W c= ].((f2 . p) - r),((f2 . p) + r).[ ) ( p1 in [#] (TOP-REAL n) & p2 in Ball (p2,r) ) by TOPGEN_5:13; hence p in W by A16, ZFMISC_1:def_2; ::_thesis: f2 .: W c= ].((f2 . p) - r),((f2 . p) + r).[ let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f2 .: W or y in ].((f2 . p) - r),((f2 . p) + r).[ ) assume y in f2 .: W ; ::_thesis: y in ].((f2 . p) - r),((f2 . p) + r).[ then consider x being Point of [:(TOP-REAL n),(TOP-REAL n):] such that A17: x in W and A18: f2 . x = y by FUNCT_2:65; consider x1, x2 being Point of (TOP-REAL n) such that A19: x = [x1,x2] by BORSUK_1:10; A20: ( f2 . (x1,x2) = x2 . i & f2 . (p1,p2) = p2 . i ) by A8; x2 in Ball (p2,r) by A17, A19, ZFMISC_1:87; then A21: |.(x2 - p2).| < r by TOPREAL9:7; dom (x2 - p2) = Seg n by FINSEQ_1:89; then (x2 . i) - (p2 . i) = (x2 - p2) . i by VALUED_1:13; then abs ((x2 . i) - (p2 . i)) <= |.(x2 - p2).| by A3, REAL_NS1:8; then abs ((f2 . x) - (f2 . p)) < r by A16, A19, A20, A21, XXREAL_0:2; hence y in ].((f2 . p) - r),((f2 . p) + r).[ by A18, RCOMP_1:1; ::_thesis: verum end; then f2 is continuous by TOPS_4:21; then consider g being Function of [:(TOP-REAL n),(TOP-REAL n):],R^1 such that A22: for p being Point of [:(TOP-REAL n),(TOP-REAL n):] for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 * r2 and A23: g is continuous by A15, JGRAPH_2:25; now__::_thesis:_for_a,_b_being_Point_of_(TOP-REAL_n)_holds_f_._(a,b)_=_g_._(a,b) let a, b be Point of (TOP-REAL n); ::_thesis: f . (a,b) = g . (a,b) A24: ( f1 . (a,b) = a . i & f2 . (a,b) = b . i ) by A7, A8; thus f . (a,b) = (a . i) * (b . i) by A6 .= g . [a,b] by A22, A24 .= g . (a,b) ; ::_thesis: verum end; hence (proj (((Seg n) --> R^1),i)) * F is continuous by A23, A4, A5, BINOP_1:2; ::_thesis: verum end; then F is continuous by WAYBEL18:6; hence TIMES n is continuous by A1, A2, YELLOW12:36; ::_thesis: verum end; suppose n is zero ; ::_thesis: TIMES n is continuous hence TIMES n is continuous by Th53; ::_thesis: verum end; end; end; end; theorem :: TOPREALC:59 for m, n being Nat for f being Function of (TOP-REAL m),(TOP-REAL n) st f is continuous holds f (-) is continuous Function of (TOP-REAL m),(TOP-REAL n) proof let m, n be Nat; ::_thesis: for f being Function of (TOP-REAL m),(TOP-REAL n) st f is continuous holds f (-) is continuous Function of (TOP-REAL m),(TOP-REAL n) let f be Function of (TOP-REAL m),(TOP-REAL n); ::_thesis: ( f is continuous implies f (-) is continuous Function of (TOP-REAL m),(TOP-REAL n) ) assume A1: f is continuous ; ::_thesis: f (-) is continuous Function of (TOP-REAL m),(TOP-REAL n) reconsider g = f (-) as Function of (TOP-REAL m),(TOP-REAL n) by Th34; for p being Point of (TOP-REAL m) for r being positive real number ex s being positive real number st g .: (Ball (p,s)) c= Ball ((g . p),r) proof let p be Point of (TOP-REAL m); ::_thesis: for r being positive real number ex s being positive real number st g .: (Ball (p,s)) c= Ball ((g . p),r) let r be positive real number ; ::_thesis: ex s being positive real number st g .: (Ball (p,s)) c= Ball ((g . p),r) consider s being positive real number such that A2: f .: (Ball ((- p),s)) c= Ball ((f . (- p)),r) by A1, TOPS_4:20; take s ; ::_thesis: g .: (Ball (p,s)) c= Ball ((g . p),r) let y be Element of (TOP-REAL n); :: according to LATTICE7:def_1 ::_thesis: ( not y in g .: (Ball (p,s)) or y in Ball ((g . p),r) ) assume y in g .: (Ball (p,s)) ; ::_thesis: y in Ball ((g . p),r) then consider x being Element of (TOP-REAL m) such that A3: x in Ball (p,s) and A4: g . x = y by FUNCT_2:65; dom g = the carrier of (TOP-REAL m) by FUNCT_2:def_1; then A5: ( g . x = f . (- x) & g . p = f . (- p) ) by VALUED_2:def_34; - x in Ball ((- p),s) by A3, Th23; then f . (- x) in f .: (Ball ((- p),s)) by FUNCT_2:35; hence y in Ball ((g . p),r) by A2, A4, A5; ::_thesis: verum end; hence f (-) is continuous Function of (TOP-REAL m),(TOP-REAL n) by TOPS_4:20; ::_thesis: verum end; registration let T be non empty TopSpace; let f be continuous Function of T,R^1; cluster - f -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = - f holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = - f implies F is continuous ) assume A1: F = - f ; ::_thesis: F is continuous consider g being Function of T,R^1 such that A2: for p being Point of T for r being real number st f . p = r holds g . p = - r and A3: g is continuous by JGRAPH_4:8; F = g proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = g . x thus F . x = - (f . x) by A1, VALUED_1:8 .= g . x by A2 ; ::_thesis: verum end; hence F is continuous by A3; ::_thesis: verum end; end; registration let T be non empty TopSpace; let f be non-empty continuous Function of T,R^1; clusterf " -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f " holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = f " implies F is continuous ) assume A1: F = f " ; ::_thesis: F is continuous dom f = the carrier of T by FUNCT_2:def_1; then for q being Point of T holds f . q <> 0 ; then consider g being Function of T,R^1 such that A2: for p being Point of T for r being real number st f . p = r holds g . p = 1 / r and A3: g is continuous by JGRAPH_2:26; F = g proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = g . x thus F . x = 1 / (f . x) by A1, VALUED_1:10 .= g . x by A2 ; ::_thesis: verum end; hence F is continuous by A3; ::_thesis: verum end; end; registration let T be non empty TopSpace; let f be continuous Function of T,R^1; let r be real number ; clusterr + f -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f + r holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = f + r implies F is continuous ) assume A1: F = f + r ; ::_thesis: F is continuous consider g being Function of T,R^1 such that A2: for p being Point of T for s being real number st f . p = s holds g . p = s + r and A3: g is continuous by JGRAPH_2:24; F = g proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = g . x thus F . x = (f . x) + r by A1, VALUED_1:2 .= g . x by A2 ; ::_thesis: verum end; hence F is continuous by A3; ::_thesis: verum end; clusterf - r -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f - r holds b1 is continuous ; clusterr (#) f -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f (#) r holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = f (#) r implies F is continuous ) assume A4: F = f (#) r ; ::_thesis: F is continuous consider g being Function of T,R^1 such that A5: for p being Point of T for s being real number st f . p = s holds g . p = r * s and A6: g is continuous by JGRAPH_2:23; F = g proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = g . x thus F . x = (f . x) * r by A4, VALUED_1:6 .= g . x by A5 ; ::_thesis: verum end; hence F is continuous by A6; ::_thesis: verum end; clusterf (/) r -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f (/) r holds b1 is continuous ; end; registration let T be non empty TopSpace; let f, g be continuous Function of T,R^1; clusterf + g -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f + g holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = f + g implies F is continuous ) assume A1: F = f + g ; ::_thesis: F is continuous consider h being Function of T,R^1 such that A2: for p being Point of T for r1, r2 being real number st f . p = r1 & g . p = r2 holds h . p = r1 + r2 and A3: h is continuous by JGRAPH_2:19; F = h proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = h . x thus F . x = (f . x) + (g . x) by A1, VALUED_1:1 .= h . x by A2 ; ::_thesis: verum end; hence F is continuous by A3; ::_thesis: verum end; clusterf - g -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f - g holds b1 is continuous proof f - g = f + (- g) ; hence for b1 being Function of T,R^1 st b1 = f - g holds b1 is continuous ; ::_thesis: verum end; clusterf (#) g -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f (#) g holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = f (#) g implies F is continuous ) assume A4: F = f (#) g ; ::_thesis: F is continuous consider h being Function of T,R^1 such that A5: for p being Point of T for r1, r2 being real number st f . p = r1 & g . p = r2 holds h . p = r1 * r2 and A6: h is continuous by JGRAPH_2:25; F = h proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = h . x thus F . x = (f . x) * (g . x) by A4, VALUED_1:5 .= h . x by A5 ; ::_thesis: verum end; hence F is continuous by A6; ::_thesis: verum end; end; registration let T be non empty TopSpace; let f be continuous Function of T,R^1; let g be non-empty continuous Function of T,R^1; clusterf /" g -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f /" g holds b1 is continuous proof f /" g = f (#) (g ") ; hence for b1 being Function of T,R^1 st b1 = f /" g holds b1 is continuous ; ::_thesis: verum end; end; registration let n be Nat; let T be non empty TopSpace; let f, g be continuous Function of T,(TOP-REAL n); clusterf <++> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <++> g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f <++> g implies h is continuous ) assume A1: h = f <++> g ; ::_thesis: h is continuous n in NAT by ORDINAL1:def_12; then consider F being Function of T,(TOP-REAL n) such that A2: for r being Point of T holds F . r = (f . r) + (g . r) and A3: F is continuous by JGRAPH_6:12; F = h proof A4: dom h = the carrier of T by FUNCT_2:def_1; let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = h . x thus F . x = (f . x) + (g . x) by A2 .= h . x by A1, A4, VALUED_2:def_45 ; ::_thesis: verum end; hence h is continuous by A3; ::_thesis: verum end; clusterf <--> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <--> g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f <--> g implies h is continuous ) assume A5: h = f <--> g ; ::_thesis: h is continuous A6: n in NAT by ORDINAL1:def_12; A7: for r being Point of T holds h . r = (f . r) - (g . r) proof let r be Point of T; ::_thesis: h . r = (f . r) - (g . r) dom h = the carrier of T by FUNCT_2:def_1; hence h . r = (f . r) - (g . r) by A5, VALUED_2:def_46; ::_thesis: verum end; for p being Point of T for V being Subset of (TOP-REAL n) st h . p in V & V is open holds ex W being Subset of T st ( p in W & W is open & h .: W c= V ) proof let p be Point of T; ::_thesis: for V being Subset of (TOP-REAL n) st h . p in V & V is open holds ex W being Subset of T st ( p in W & W is open & h .: W c= V ) let V be Subset of (TOP-REAL n); ::_thesis: ( h . p in V & V is open implies ex W being Subset of T st ( p in W & W is open & h .: W c= V ) ) assume ( h . p in V & V is open ) ; ::_thesis: ex W being Subset of T st ( p in W & W is open & h .: W c= V ) then A8: h . p in Int V by TOPS_1:23; reconsider r = h . p as Point of (Euclid n) by EUCLID:67; consider r0 being real number such that A9: ( r0 > 0 & Ball (r,r0) c= V ) by A8, GOBOARD6:5; reconsider r01 = f . p as Point of (Euclid n) by EUCLID:67; reconsider G1 = Ball (r01,(r0 / 2)) as Subset of (TOP-REAL n) by EUCLID:67; A10: f . p in G1 by A9, GOBOARD6:1; A11: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8; G1 is open by A11, TOPMETR:14, TOPS_3:76; then consider W1 being Subset of T such that A12: ( p in W1 & W1 is open & f .: W1 c= G1 ) by A10, JGRAPH_2:10; reconsider r02 = g . p as Point of (Euclid n) by EUCLID:67; reconsider G2 = Ball (r02,(r0 / 2)) as Subset of (TOP-REAL n) by EUCLID:67; A13: g . p in G2 by A9, GOBOARD6:1; G2 is open by A11, TOPMETR:14, TOPS_3:76; then consider W2 being Subset of T such that A14: ( p in W2 & W2 is open & g .: W2 c= G2 ) by A13, JGRAPH_2:10; take W = W1 /\ W2; ::_thesis: ( p in W & W is open & h .: W c= V ) thus p in W by A12, A14, XBOOLE_0:def_4; ::_thesis: ( W is open & h .: W c= V ) thus W is open by A12, A14; ::_thesis: h .: W c= V let x be Element of (TOP-REAL n); :: according to LATTICE7:def_1 ::_thesis: ( not x in h .: W or x in V ) assume x in h .: W ; ::_thesis: x in V then consider z being set such that A15: ( z in dom h & z in W & h . z = x ) by FUNCT_1:def_6; A16: z in W1 by A15, XBOOLE_0:def_4; reconsider pz = z as Point of T by A15; dom f = the carrier of T by FUNCT_2:def_1; then A17: f . pz in f .: W1 by A16, FUNCT_1:def_6; reconsider bb1 = f . pz as Point of (Euclid n) by EUCLID:67; dist (r01,bb1) < r0 / 2 by A12, A17, METRIC_1:11; then A18: |.((f . p) - (f . pz)).| < r0 / 2 by A6, JGRAPH_1:28; A19: z in W2 by A15, XBOOLE_0:def_4; dom g = the carrier of T by FUNCT_2:def_1; then A20: g . pz in g .: W2 by A19, FUNCT_1:def_6; reconsider bb2 = g . pz as Point of (Euclid n) by EUCLID:67; dist (r02,bb2) < r0 / 2 by A14, A20, METRIC_1:11; then A21: |.((g . p) - (g . pz)).| < r0 / 2 by A6, JGRAPH_1:28; A22: (f . pz) - (g . pz) = x by A7, A15; reconsider bb0 = (f . pz) - (g . pz) as Point of (Euclid n) by EUCLID:67; A23: h . pz = (f . pz) - (g . pz) by A7; ((f . p) - (g . p)) - ((f . pz) - (g . pz)) = (((f . p) - (g . p)) - (f . pz)) + (g . pz) by EUCLID:47 .= (((f . p) + (- (g . p))) + (- (f . pz))) + (g . pz) .= (((f . p) + (- (f . pz))) + (- (g . p))) + (g . pz) by EUCLID:26 .= (((f . p) - (f . pz)) - (g . p)) + (g . pz) .= (((f . p) - (f . pz)) + (g . pz)) + (- (g . p)) by EUCLID:26 .= ((f . p) - (f . pz)) + ((g . pz) - (g . p)) by EUCLID:26 ; then A24: |.(((f . p) - (g . p)) - ((f . pz) - (g . pz))).| <= |.((f . p) - (f . pz)).| + |.((g . pz) - (g . p)).| by A6, TOPRNS_1:29; A25: |.((g . p) - (g . pz)).| = |.(- ((g . pz) - (g . p))).| by EUCLID:44 .= |.((g . pz) - (g . p)).| by A6, TOPRNS_1:26 ; |.((f . p) - (f . pz)).| + |.((g . p) - (g . pz)).| < (r0 / 2) + (r0 / 2) by A18, A21, XREAL_1:8; then |.(((f . p) - (g . p)) - ((f . pz) - (g . pz))).| < r0 by A24, A25, XXREAL_0:2; then |.((h . p) - (h . pz)).| < r0 by A7, A23; then dist (r,bb0) < r0 by A6, A15, A22, JGRAPH_1:28; then x in Ball (r,r0) by A22, METRIC_1:11; hence x in V by A9; ::_thesis: verum end; hence h is continuous by JGRAPH_2:10; ::_thesis: verum end; clusterf <##> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <##> g holds b1 is continuous proof A26: f <##> g = (TIMES n) .: (f,g) by Th54; <:f,g:> is continuous Function of T,[:(TOP-REAL n),(TOP-REAL n):] by YELLOW12:41; hence for b1 being Function of T,(TOP-REAL n) st b1 = f <##> g holds b1 is continuous by A26; ::_thesis: verum end; end; registration let n be Nat; let T be non empty TopSpace; let f be continuous Function of T,(TOP-REAL n); let g be continuous Function of T,R^1; set I = incl (g,n); clusterf <+> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <+> g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f <+> g implies h is continuous ) assume A1: h = f <+> g ; ::_thesis: h is continuous reconsider G = f <++> (incl (g,n)) as Function of T,(TOP-REAL n) by Th39; h = G by A1, Th49; hence h is continuous ; ::_thesis: verum end; clusterf <-> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <-> g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f <-> g implies h is continuous ) assume A2: h = f <-> g ; ::_thesis: h is continuous reconsider G = f <--> (incl (g,n)) as Function of T,(TOP-REAL n) by Th40; h = G by A2, Th50; hence h is continuous ; ::_thesis: verum end; clusterf <#> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <#> g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f <#> g implies h is continuous ) assume A3: h = f <#> g ; ::_thesis: h is continuous reconsider G = f <##> (incl (g,n)) as Function of T,(TOP-REAL n) by Th41; h = G by A3, Th51; hence h is continuous ; ::_thesis: verum end; end; registration let n be Nat; let T be non empty TopSpace; let f be continuous Function of T,(TOP-REAL n); let g be non-empty continuous Function of T,R^1; clusterf g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f g implies h is continuous ) reconsider g1 = g " as Function of T,R^1 ; g1 is continuous ; hence ( h = f g implies h is continuous ) ; ::_thesis: verum end; end; registration let n be Nat; let T be non empty TopSpace; let r be real number ; let f be continuous Function of T,(TOP-REAL n); clusterf [+] r -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f [+] r holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f [+] r implies h is continuous ) assume A1: h = f [+] r ; ::_thesis: h is continuous reconsider r = r as Element of R^1 by XREAL_0:def_1; dom f = the carrier of T by FUNCT_2:def_1; then h = f <+> (T --> r) by A1, Th29; hence h is continuous ; ::_thesis: verum end; clusterf [-] r -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f [-] r holds b1 is continuous ; clusterf [#] r -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f [#] r holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f [#] r implies h is continuous ) assume A2: h = f [#] r ; ::_thesis: h is continuous reconsider r = r as Element of R^1 by XREAL_0:def_1; dom f = the carrier of T by FUNCT_2:def_1; then h = f <#> (T --> r) by A2, Th31; hence h is continuous ; ::_thesis: verum end; clusterf [/] r -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f [/] r holds b1 is continuous ; end; theorem Th60: :: TOPREALC:60 for r being non negative real number for n being non zero Nat for p being Point of (Tcircle ((0. (TOP-REAL n)),r)) holds - p is Point of (Tcircle ((0. (TOP-REAL n)),r)) proof let r be non negative real number ; ::_thesis: for n being non zero Nat for p being Point of (Tcircle ((0. (TOP-REAL n)),r)) holds - p is Point of (Tcircle ((0. (TOP-REAL n)),r)) let n be non zero Nat; ::_thesis: for p being Point of (Tcircle ((0. (TOP-REAL n)),r)) holds - p is Point of (Tcircle ((0. (TOP-REAL n)),r)) let p be Point of (Tcircle ((0. (TOP-REAL n)),r)); ::_thesis: - p is Point of (Tcircle ((0. (TOP-REAL n)),r)) reconsider p1 = p as Point of (TOP-REAL n) by PRE_TOPC:25; A1: n in NAT by ORDINAL1:def_12; then A2: the carrier of (Tcircle ((0. (TOP-REAL n)),r)) = Sphere ((0. (TOP-REAL n)),r) by TOPREALB:9; |.((- p1) - (0. (TOP-REAL n))).| = |.(- p1).| by RLVECT_1:13 .= |.p1.| by EUCLID:71 .= |.(p1 - (0. (TOP-REAL n))).| by RLVECT_1:13 .= r by A1, A2, TOPREAL9:9 ; hence - p is Point of (Tcircle ((0. (TOP-REAL n)),r)) by A2, A1, TOPREAL9:9; ::_thesis: verum end; theorem Th61: :: TOPREALC:61 for n being Nat for r being non negative real number for f being Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) holds f (-) is Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) proof let n be Nat; ::_thesis: for r being non negative real number for f being Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) holds f (-) is Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) let r be non negative real number ; ::_thesis: for f being Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) holds f (-) is Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) set X = the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)); let f be Function of the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n); ::_thesis: f (-) is Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) set g = f (-) ; A1: dom (f (-)) = dom f by VALUED_2:def_34; A2: dom f = the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) by FUNCT_2:def_1; for x being set st x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) holds (f (-)) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) implies (f (-)) . x in the carrier of (TOP-REAL n) ) assume x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) ; ::_thesis: (f (-)) . x in the carrier of (TOP-REAL n) then reconsider x = x as Element of the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) ; reconsider y = - x as Element of the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) by Th60; reconsider f = f as Function of the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) ; (f (-)) . x = f . y by A1, A2, VALUED_2:def_34; hence (f (-)) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f (-) is Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) by A1, A2, FUNCT_2:3; ::_thesis: verum end; definition let n be Nat; let r be non negative real number ; let X be Subset of (Tcircle ((0. (TOP-REAL (n + 1))),r)); :: original: (-) redefine func (-) X -> Subset of (Tcircle ((0. (TOP-REAL (n + 1))),r)); coherence (-) X is Subset of (Tcircle ((0. (TOP-REAL (n + 1))),r)) proof set T = Tcircle ((0. (TOP-REAL (n + 1))),r); (-) X c= the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (-) X or x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) ) assume A1: x in (-) X ; ::_thesis: x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) then reconsider x = x as Element of (-) X ; - x in X by A1, Th24; then - (- x) is Point of (Tcircle ((0. (TOP-REAL (n + 1))),r)) by Th60; hence x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) ; ::_thesis: verum end; hence (-) X is Subset of (Tcircle ((0. (TOP-REAL (n + 1))),r)) ; ::_thesis: verum end; end; registration let m be Nat; let r be non negative real number ; let X be open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)); cluster (-) X -> open for Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)); coherence for b1 being Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) st b1 = (-) X holds b1 is open proof set T = Tcircle ((0. (TOP-REAL (m + 1))),r); ex G being Subset of (TOP-REAL (m + 1)) st ( G is open & (-) X = G /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) ) proof consider G being Subset of (TOP-REAL (m + 1)) such that A1: G is open and A2: X = G /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by TSP_1:def_1; take (-) G ; ::_thesis: ( (-) G is open & (-) X = ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) ) thus (-) G is open by A1; ::_thesis: (-) X = ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) thus (-) X c= ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) :: according to XBOOLE_0:def_10 ::_thesis: ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) c= (-) X proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (-) X or x in ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) ) assume A3: x in (-) X ; ::_thesis: x in ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) then reconsider x = x as Element of (-) X ; - x in X by A3, Th24; then - x in G by A2, XBOOLE_0:def_4; then x in (-) G by Th24; hence x in ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by A3, XBOOLE_0:def_4; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) or x in (-) X ) assume A4: x in ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) ; ::_thesis: x in (-) X then reconsider x = x as Element of (-) G by XBOOLE_0:def_4; x in (-) G by A4, XBOOLE_0:def_4; then A5: - x in G by Th24; x in the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by A4, XBOOLE_0:def_4; then - x is Point of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by Th60; then - x in X by A2, A5, XBOOLE_0:def_4; hence x in (-) X by Th24; ::_thesis: verum end; hence for b1 being Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) st b1 = (-) X holds b1 is open by TSP_1:def_1; ::_thesis: verum end; end; theorem :: TOPREALC:62 for m being Nat for r being non negative real number for f being continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) holds f (-) is continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) proof let m be Nat; ::_thesis: for r being non negative real number for f being continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) holds f (-) is continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) let r be non negative real number ; ::_thesis: for f being continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) holds f (-) is continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) set T = Tcircle ((0. (TOP-REAL (m + 1))),r); let f be continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m); ::_thesis: f (-) is continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) reconsider g = f (-) as Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) by Th61; for p being Point of (Tcircle ((0. (TOP-REAL (m + 1))),r)) for r being positive real number ex W being open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) st ( p in W & g .: W c= Ball ((g . p),r) ) proof let p be Point of (Tcircle ((0. (TOP-REAL (m + 1))),r)); ::_thesis: for r being positive real number ex W being open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) st ( p in W & g .: W c= Ball ((g . p),r) ) let r be positive real number ; ::_thesis: ex W being open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) st ( p in W & g .: W c= Ball ((g . p),r) ) reconsider q = - p as Point of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by Th60; consider W being open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) such that A1: q in W and A2: f .: W c= Ball ((f . q),r) by TOPS_4:18; reconsider W1 = (-) W as open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) ; take W1 ; ::_thesis: ( p in W1 & g .: W1 c= Ball ((g . p),r) ) - q in W1 by A1, Def3; hence p in W1 ; ::_thesis: g .: W1 c= Ball ((g . p),r) let y be Element of (TOP-REAL m); :: according to LATTICE7:def_1 ::_thesis: ( not y in g .: W1 or y in Ball ((g . p),r) ) assume y in g .: W1 ; ::_thesis: y in Ball ((g . p),r) then consider x being Element of (Tcircle ((0. (TOP-REAL (m + 1))),r)) such that A3: x in W1 and A4: g . x = y by FUNCT_2:65; dom g = the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by FUNCT_2:def_1; then A5: ( g . x = f . (- x) & g . p = f . (- p) ) by VALUED_2:def_34; - x in (-) W1 by A3, Def3; then f . (- x) in f .: W by FUNCT_2:35; hence y in Ball ((g . p),r) by A2, A4, A5; ::_thesis: verum end; hence f (-) is continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) by TOPS_4:18; ::_thesis: verum end;
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: :: TOPREALC semantic presentation begin theorem :: TOPREALC:1 for X being trivial set for Y being set st X,Y are_equipotent holds Y is trivial proof let X be trivial set ; ::_thesis: for Y being set st X,Y are_equipotent holds Y is trivial let Y be set ; ::_thesis: ( X,Y are_equipotent implies Y is trivial ) assume A1: X,Y are_equipotent ; ::_thesis: Y is trivial A2: card X < 2 by NAT_D:60; A3: Y is finite by A1, CARD_1:38; card X = card Y by A1, CARD_1:5; hence Y is trivial by A2, A3, NAT_D:60; ::_thesis: verum end; registration let r be real number ; clusterr ^2 -> non negative ; coherence not r ^2 is negative ; end; registration let r be positive real number ; clusterr ^2 -> positive ; coherence r ^2 is positive ; end; registration cluster sqrt 0 -> zero ; coherence sqrt 0 is empty by SQUARE_1:17; end; registration let f be empty set ; clusterf ^2 -> empty ; coherence sqr f is empty ; cluster|.f.| -> zero ; coherence |.f.| is empty by RVSUM_1:72; end; theorem Th2: :: TOPREALC:2 for c1, c2 being complex number for f being complex-valued Function holds f (#) (c1 + c2) = (f (#) c1) + (f (#) c2) proof let c1, c2 be complex number ; ::_thesis: for f being complex-valued Function holds f (#) (c1 + c2) = (f (#) c1) + (f (#) c2) let f be complex-valued Function; ::_thesis: f (#) (c1 + c2) = (f (#) c1) + (f (#) c2) A1: dom (f (#) (c1 + c2)) = dom f by VALUED_1:def_5; A2: dom (f (#) c1) = dom f by VALUED_1:def_5; A3: dom (f (#) c2) = dom f by VALUED_1:def_5; A4: dom ((f (#) c1) + (f (#) c2)) = (dom (f (#) c1)) /\ (dom (f (#) c2)) by VALUED_1:def_1; hence dom (f (#) (c1 + c2)) = dom ((f (#) c1) + (f (#) c2)) by A2, A3, VALUED_1:def_5; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom (f (#) (c1 + c2)) or (f (#) (c1 + c2)) . b1 = ((f (#) c1) + (f (#) c2)) . b1 ) let x be set ; ::_thesis: ( not x in dom (f (#) (c1 + c2)) or (f (#) (c1 + c2)) . x = ((f (#) c1) + (f (#) c2)) . x ) assume A5: x in dom (f (#) (c1 + c2)) ; ::_thesis: (f (#) (c1 + c2)) . x = ((f (#) c1) + (f (#) c2)) . x hence (f (#) (c1 + c2)) . x = (f . x) * (c1 + c2) by VALUED_1:def_5 .= ((f . x) * c1) + ((f . x) * c2) .= ((f (#) c1) . x) + ((f . x) * c2) by A1, A2, A5, VALUED_1:def_5 .= ((f (#) c1) . x) + ((f (#) c2) . x) by A1, A3, A5, VALUED_1:def_5 .= ((f (#) c1) + (f (#) c2)) . x by A1, A2, A3, A4, A5, VALUED_1:def_1 ; ::_thesis: verum end; theorem :: TOPREALC:3 for c1, c2 being complex number for f being complex-valued Function holds f (#) (c1 - c2) = (f (#) c1) - (f (#) c2) proof let c1, c2 be complex number ; ::_thesis: for f being complex-valued Function holds f (#) (c1 - c2) = (f (#) c1) - (f (#) c2) let f be complex-valued Function; ::_thesis: f (#) (c1 - c2) = (f (#) c1) - (f (#) c2) thus f (#) (c1 - c2) = (f (#) c1) + (f (#) (- c2)) by Th2 .= (f (#) c1) - (f (#) c2) by VALUED_2:24 ; ::_thesis: verum end; theorem Th4: :: TOPREALC:4 for c being complex number for f, g being complex-valued Function holds (f (/) c) + (g (/) c) = (f + g) (/) c proof let c be complex number ; ::_thesis: for f, g being complex-valued Function holds (f (/) c) + (g (/) c) = (f + g) (/) c let f, g be complex-valued Function; ::_thesis: (f (/) c) + (g (/) c) = (f + g) (/) c A1: dom ((f (/) c) + (g (/) c)) = (dom (f (/) c)) /\ (dom (g (/) c)) by VALUED_1:def_1; A2: dom (f (/) c) = dom f by VALUED_2:28; A3: dom (g (/) c) = dom g by VALUED_2:28; A4: dom ((f + g) (/) c) = dom (f + g) by VALUED_2:28; A5: dom (f + g) = (dom f) /\ (dom g) by VALUED_1:def_1; thus dom ((f (/) c) + (g (/) c)) = dom ((f + g) (/) c) by A1, A2, A3, A4, VALUED_1:def_1; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f (/) c) + (g (/) c)) or ((f (/) c) + (g (/) c)) . b1 = ((f + g) (/) c) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f (/) c) + (g (/) c)) or ((f (/) c) + (g (/) c)) . x = ((f + g) (/) c) . x ) assume A6: x in dom ((f (/) c) + (g (/) c)) ; ::_thesis: ((f (/) c) + (g (/) c)) . x = ((f + g) (/) c) . x thus ((f + g) (/) c) . x = ((f + g) . x) / c by VALUED_2:29 .= ((f . x) + (g . x)) / c by A1, A2, A3, A6, A5, VALUED_1:def_1 .= ((f . x) / c) + ((g . x) / c) .= ((f . x) / c) + ((g (/) c) . x) by VALUED_2:29 .= ((f (/) c) . x) + ((g (/) c) . x) by VALUED_2:29 .= ((f (/) c) + (g (/) c)) . x by A6, VALUED_1:def_1 ; ::_thesis: verum end; theorem :: TOPREALC:5 for c being complex number for f, g being complex-valued Function holds (f (/) c) - (g (/) c) = (f - g) (/) c proof let c be complex number ; ::_thesis: for f, g being complex-valued Function holds (f (/) c) - (g (/) c) = (f - g) (/) c let f, g be complex-valued Function; ::_thesis: (f (/) c) - (g (/) c) = (f - g) (/) c thus (f (/) c) - (g (/) c) = (f (/) c) + ((- g) (/) c) by VALUED_2:30 .= (f - g) (/) c by Th4 ; ::_thesis: verum end; theorem :: TOPREALC:6 for c1, c2 being complex number for f, g being complex-valued Function st c1 <> 0 & c2 <> 0 holds (f (/) c1) - (g (/) c2) = ((f (#) c2) - (g (#) c1)) (/) (c1 * c2) proof let c1, c2 be complex number ; ::_thesis: for f, g being complex-valued Function st c1 <> 0 & c2 <> 0 holds (f (/) c1) - (g (/) c2) = ((f (#) c2) - (g (#) c1)) (/) (c1 * c2) let f, g be complex-valued Function; ::_thesis: ( c1 <> 0 & c2 <> 0 implies (f (/) c1) - (g (/) c2) = ((f (#) c2) - (g (#) c1)) (/) (c1 * c2) ) assume A1: ( c1 <> 0 & c2 <> 0 ) ; ::_thesis: (f (/) c1) - (g (/) c2) = ((f (#) c2) - (g (#) c1)) (/) (c1 * c2) A2: dom (f (/) c1) = dom f by VALUED_2:28; A3: dom (g (/) c2) = dom g by VALUED_2:28; A4: dom (f (#) c2) = dom f by VALUED_1:def_5; A5: dom (g (#) c1) = dom g by VALUED_1:def_5; A6: dom ((f (/) c1) - (g (/) c2)) = (dom (f (/) c1)) /\ (dom (g (/) c2)) by VALUED_1:12; A7: dom ((f (#) c2) - (g (#) c1)) = (dom (f (#) c2)) /\ (dom (g (#) c1)) by VALUED_1:12; hence dom ((f (/) c1) - (g (/) c2)) = dom (((f (#) c2) - (g (#) c1)) (/) (c1 * c2)) by A2, A3, A4, A5, A6, VALUED_2:28; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f (/) c1) - (g (/) c2)) or ((f (/) c1) - (g (/) c2)) . b1 = (((f (#) c2) - (g (#) c1)) (/) (c1 * c2)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f (/) c1) - (g (/) c2)) or ((f (/) c1) - (g (/) c2)) . x = (((f (#) c2) - (g (#) c1)) (/) (c1 * c2)) . x ) assume A8: x in dom ((f (/) c1) - (g (/) c2)) ; ::_thesis: ((f (/) c1) - (g (/) c2)) . x = (((f (#) c2) - (g (#) c1)) (/) (c1 * c2)) . x hence ((f (/) c1) - (g (/) c2)) . x = ((f (/) c1) . x) - ((g (/) c2) . x) by VALUED_1:13 .= ((f . x) / c1) - ((g (/) c2) . x) by VALUED_2:29 .= ((f . x) / c1) - ((g . x) / c2) by VALUED_2:29 .= (((f . x) * c2) - ((g . x) * c1)) / (c1 * c2) by A1, XCMPLX_1:130 .= (((f . x) * c2) - ((g (#) c1) . x)) / (c1 * c2) by VALUED_1:6 .= (((f (#) c2) . x) - ((g (#) c1) . x)) / (c1 * c2) by VALUED_1:6 .= (((f (#) c2) - (g (#) c1)) . x) / (c1 * c2) by A2, A3, A4, A5, A6, A7, A8, VALUED_1:13 .= (((f (#) c2) - (g (#) c1)) (/) (c1 * c2)) . x by VALUED_2:29 ; ::_thesis: verum end; theorem :: TOPREALC:7 for c being complex number for f, g being complex-valued Function st c <> 0 holds (f (/) c) - g = (f - (c (#) g)) (/) c proof let c be complex number ; ::_thesis: for f, g being complex-valued Function st c <> 0 holds (f (/) c) - g = (f - (c (#) g)) (/) c let f, g be complex-valued Function; ::_thesis: ( c <> 0 implies (f (/) c) - g = (f - (c (#) g)) (/) c ) assume A1: c <> 0 ; ::_thesis: (f (/) c) - g = (f - (c (#) g)) (/) c A2: dom ((f (/) c) - g) = (dom (f (/) c)) /\ (dom g) by VALUED_1:12 .= (dom f) /\ (dom g) by VALUED_2:28 ; A3: dom (f - (c (#) g)) = (dom f) /\ (dom (c (#) g)) by VALUED_1:12 .= (dom f) /\ (dom g) by VALUED_1:def_5 ; hence dom ((f (/) c) - g) = dom ((f - (c (#) g)) (/) c) by A2, VALUED_2:28; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f (/) c) - g) or ((f (/) c) - g) . b1 = ((f - (c (#) g)) (/) c) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f (/) c) - g) or ((f (/) c) - g) . x = ((f - (c (#) g)) (/) c) . x ) assume A4: x in dom ((f (/) c) - g) ; ::_thesis: ((f (/) c) - g) . x = ((f - (c (#) g)) (/) c) . x hence ((f (/) c) - g) . x = ((f (/) c) . x) - (g . x) by VALUED_1:13 .= ((f . x) / c) - (g . x) by VALUED_2:29 .= ((f . x) / c) - ((c * (g . x)) / c) by A1, XCMPLX_1:89 .= ((f . x) / c) - (((c (#) g) . x) / c) by VALUED_1:6 .= ((f . x) - ((c (#) g) . x)) / c .= ((f - (c (#) g)) . x) / c by A2, A3, A4, VALUED_1:13 .= ((f - (c (#) g)) (/) c) . x by VALUED_2:29 ; ::_thesis: verum end; theorem :: TOPREALC:8 for c, d being complex number for f being complex-valued Function holds (c - d) (#) f = (c (#) f) - (d (#) f) proof let c, d be complex number ; ::_thesis: for f being complex-valued Function holds (c - d) (#) f = (c (#) f) - (d (#) f) let f be complex-valued Function; ::_thesis: (c - d) (#) f = (c (#) f) - (d (#) f) dom ((c (#) f) - (d (#) f)) = (dom (c (#) f)) /\ (dom (d (#) f)) by VALUED_1:12 .= (dom f) /\ (dom (d (#) f)) by VALUED_1:def_5 .= (dom f) /\ (dom f) by VALUED_1:def_5 ; hence A1: dom ((c - d) (#) f) = dom ((c (#) f) - (d (#) f)) by VALUED_1:def_5; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((c - d) (#) f) or ((c - d) (#) f) . b1 = ((c (#) f) - (d (#) f)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((c - d) (#) f) or ((c - d) (#) f) . x = ((c (#) f) - (d (#) f)) . x ) assume A2: x in dom ((c - d) (#) f) ; ::_thesis: ((c - d) (#) f) . x = ((c (#) f) - (d (#) f)) . x thus ((c - d) (#) f) . x = (c - d) * (f . x) by VALUED_1:6 .= (c * (f . x)) - (d * (f . x)) .= (c * (f . x)) - ((d (#) f) . x) by VALUED_1:6 .= ((c (#) f) . x) - ((d (#) f) . x) by VALUED_1:6 .= ((c (#) f) - (d (#) f)) . x by A1, A2, VALUED_1:13 ; ::_thesis: verum end; theorem :: TOPREALC:9 for f, g being complex-valued Function holds (f - g) ^2 = (g - f) ^2 proof let f, g be complex-valued Function; ::_thesis: (f - g) ^2 = (g - f) ^2 A1: dom (f - g) = (dom f) /\ (dom g) by VALUED_1:12; A2: dom (g - f) = (dom g) /\ (dom f) by VALUED_1:12; A3: dom ((f - g) ^2) = dom (f - g) by VALUED_1:11; hence dom ((f - g) ^2) = dom ((g - f) ^2) by A1, A2, VALUED_1:11; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f - g) ^2) or ((f - g) ^2) . b1 = ((g - f) ^2) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f - g) ^2) or ((f - g) ^2) . x = ((g - f) ^2) . x ) assume A4: x in dom ((f - g) ^2) ; ::_thesis: ((f - g) ^2) . x = ((g - f) ^2) . x then A5: (f - g) . x = (f . x) - (g . x) by A3, VALUED_1:13; A6: (g - f) . x = (g . x) - (f . x) by A1, A3, A4, A2, VALUED_1:13; thus ((f - g) ^2) . x = ((f - g) . x) * ((f - g) . x) by VALUED_1:5 .= ((g - f) . x) * ((g - f) . x) by A5, A6 .= ((g - f) ^2) . x by VALUED_1:5 ; ::_thesis: verum end; theorem :: TOPREALC:10 for c being complex number for f being complex-valued Function holds (f (/) c) ^2 = (f ^2) (/) (c ^2) proof let c be complex number ; ::_thesis: for f being complex-valued Function holds (f (/) c) ^2 = (f ^2) (/) (c ^2) let f be complex-valued Function; ::_thesis: (f (/) c) ^2 = (f ^2) (/) (c ^2) thus dom ((f (/) c) ^2) = dom (f (/) c) by VALUED_1:11 .= dom f by VALUED_2:28 .= dom (f ^2) by VALUED_1:11 .= dom ((f ^2) (/) (c ^2)) by VALUED_2:28 ; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f (/) c) ^2) or ((f (/) c) ^2) . b1 = ((f ^2) (/) (c ^2)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f (/) c) ^2) or ((f (/) c) ^2) . x = ((f ^2) (/) (c ^2)) . x ) assume x in dom ((f (/) c) ^2) ; ::_thesis: ((f (/) c) ^2) . x = ((f ^2) (/) (c ^2)) . x thus ((f (/) c) ^2) . x = ((f (/) c) . x) ^2 by VALUED_1:11 .= ((f . x) / c) ^2 by VALUED_2:29 .= ((f . x) ^2) / (c ^2) by XCMPLX_1:76 .= ((f ^2) . x) / (c ^2) by VALUED_1:11 .= ((f ^2) (/) (c ^2)) . x by VALUED_2:29 ; ::_thesis: verum end; theorem Th11: :: TOPREALC:11 for n being Nat for r, s being real number holds |.((n |-> r) - (n |-> s)).| = (sqrt n) * (abs (r - s)) proof let n be Nat; ::_thesis: for r, s being real number holds |.((n |-> r) - (n |-> s)).| = (sqrt n) * (abs (r - s)) let r, s be real number ; ::_thesis: |.((n |-> r) - (n |-> s)).| = (sqrt n) * (abs (r - s)) thus |.((n |-> r) - (n |-> s)).| = sqrt (Sum (sqr (n |-> (r - s)))) by RVSUM_1:30 .= sqrt (Sum (n |-> ((r - s) ^2))) by RVSUM_1:56 .= sqrt (n * ((r - s) ^2)) by RVSUM_1:80 .= (sqrt n) * (sqrt ((r - s) ^2)) by SQUARE_1:29 .= (sqrt n) * (abs (r - s)) by COMPLEX1:72 ; ::_thesis: verum end; registration let f be complex-valued Function; let x be set ; let c be complex number ; clusterf +* (x,c) -> complex-valued ; coherence f +* (x,c) is complex-valued proof set F = f +* (x,c); let a be set ; :: according to VALUED_0:def_7 ::_thesis: ( not a in dom (f +* (x,c)) or (f +* (x,c)) . a is complex ) assume a in dom (f +* (x,c)) ; ::_thesis: (f +* (x,c)) . a is complex percases ( ( x in dom f & x = a ) or ( x in dom f & x <> a ) or not x in dom f ) ; suppose ( x in dom f & x = a ) ; ::_thesis: (f +* (x,c)) . a is complex hence (f +* (x,c)) . a is complex by FUNCT_7:31; ::_thesis: verum end; suppose ( x in dom f & x <> a ) ; ::_thesis: (f +* (x,c)) . a is complex then (f +* (x,c)) . a = f . a by FUNCT_7:32; hence (f +* (x,c)) . a is complex ; ::_thesis: verum end; suppose not x in dom f ; ::_thesis: (f +* (x,c)) . a is complex then (f +* (x,c)) . a = f . a by FUNCT_7:def_3; hence (f +* (x,c)) . a is complex ; ::_thesis: verum end; end; end; end; theorem Th12: :: TOPREALC:12 for x being set for n being Nat for c being complex number holds ((0* n) +* (x,c)) ^2 = (0* n) +* (x,(c ^2)) proof let x be set ; ::_thesis: for n being Nat for c being complex number holds ((0* n) +* (x,c)) ^2 = (0* n) +* (x,(c ^2)) let n be Nat; ::_thesis: for c being complex number holds ((0* n) +* (x,c)) ^2 = (0* n) +* (x,(c ^2)) let c be complex number ; ::_thesis: ((0* n) +* (x,c)) ^2 = (0* n) +* (x,(c ^2)) set f = (0* n) +* (x,c); set g = (0* n) +* (x,(c ^2)); A1: dom ((0* n) +* (x,c)) = dom (0* n) by FUNCT_7:30; A2: dom ((0* n) +* (x,(c ^2))) = dom (0* n) by FUNCT_7:30; A3: dom (((0* n) +* (x,c)) ^2) = dom ((0* n) +* (x,c)) by VALUED_1:11; thus dom (((0* n) +* (x,c)) ^2) = dom ((0* n) +* (x,(c ^2))) by A1, A2, VALUED_1:11; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom (((0* n) +* (x,c)) ^2) or (((0* n) +* (x,c)) ^2) . b1 = ((0* n) +* (x,(c ^2))) . b1 ) let a be set ; ::_thesis: ( not a in dom (((0* n) +* (x,c)) ^2) or (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a ) assume A4: a in dom (((0* n) +* (x,c)) ^2) ; ::_thesis: (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a A5: (((0* n) +* (x,c)) ^2) . a = (((0* n) +* (x,c)) . a) ^2 by VALUED_1:11; percases ( a = x or a <> x ) ; supposeA6: a = x ; ::_thesis: (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a then ((0* n) +* (x,c)) . a = c by A1, A3, A4, FUNCT_7:31; hence (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a by A6, A1, A3, A4, A5, FUNCT_7:31; ::_thesis: verum end; supposeA7: a <> x ; ::_thesis: (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a then A8: ((0* n) +* (x,c)) . a = (n |-> 0) . a by FUNCT_7:32 .= {} . x ; ((0* n) +* (x,(c ^2))) . a = (n |-> 0) . a by A7, FUNCT_7:32 .= {} . x ; hence (((0* n) +* (x,c)) ^2) . a = ((0* n) +* (x,(c ^2))) . a by A5, A8; ::_thesis: verum end; end; end; theorem Th13: :: TOPREALC:13 for x being set for n being Nat for r being real number st x in Seg n holds |.((0* n) +* (x,r)).| = abs r proof let x be set ; ::_thesis: for n being Nat for r being real number st x in Seg n holds |.((0* n) +* (x,r)).| = abs r let n be Nat; ::_thesis: for r being real number st x in Seg n holds |.((0* n) +* (x,r)).| = abs r let r be real number ; ::_thesis: ( x in Seg n implies |.((0* n) +* (x,r)).| = abs r ) set f = (0* n) +* (x,r); A1: r ^2 in REAL by XREAL_0:def_1; A2: n in NAT by ORDINAL1:def_12; assume A3: x in Seg n ; ::_thesis: |.((0* n) +* (x,r)).| = abs r ((0* n) +* (x,r)) ^2 = (0* n) +* (x,(r ^2)) by Th12; then Sum (((0* n) +* (x,r)) ^2) = r ^2 by A3, A1, A2, JORDAN2B:10; hence |.((0* n) +* (x,r)).| = abs r by COMPLEX1:72; ::_thesis: verum end; theorem Th14: :: TOPREALC:14 for x being set for n being Nat holds (0.REAL n) +* (x,0) = 0.REAL n proof let x be set ; ::_thesis: for n being Nat holds (0.REAL n) +* (x,0) = 0.REAL n let n be Nat; ::_thesis: (0.REAL n) +* (x,0) = 0.REAL n set p = (0.REAL n) +* (x,0); A1: dom ((0.REAL n) +* (x,0)) = Seg n by FINSEQ_1:89; A2: dom ((0.REAL n) +* (x,0)) = dom (0.REAL n) by FUNCT_7:30; A3: dom (0.REAL n) = Seg n by FINSEQ_1:89; now__::_thesis:_for_z_being_set_st_z_in_dom_((0.REAL_n)_+*_(x,0))_holds_ ((0.REAL_n)_+*_(x,0))_._z_=_(0.REAL_n)_._z let z be set ; ::_thesis: ( z in dom ((0.REAL n) +* (x,0)) implies ((0.REAL n) +* (x,0)) . b1 = (0.REAL n) . b1 ) assume A4: z in dom ((0.REAL n) +* (x,0)) ; ::_thesis: ((0.REAL n) +* (x,0)) . b1 = (0.REAL n) . b1 percases ( z = x or z <> x ) ; suppose z = x ; ::_thesis: ((0.REAL n) +* (x,0)) . b1 = (0.REAL n) . b1 hence ((0.REAL n) +* (x,0)) . z = 0 by A1, A3, A4, FUNCT_7:31 .= (0.REAL n) . z ; ::_thesis: verum end; suppose z <> x ; ::_thesis: ((0.REAL n) +* (x,0)) . b1 = (0.REAL n) . b1 hence ((0.REAL n) +* (x,0)) . z = (0.REAL n) . z by FUNCT_7:32; ::_thesis: verum end; end; end; hence (0.REAL n) +* (x,0) = 0.REAL n by A2, FUNCT_1:2; ::_thesis: verum end; theorem Th15: :: TOPREALC:15 for x being set for n being Nat for r being real number for f1 being real-valued b2 -element FinSequence holds mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) proof let x be set ; ::_thesis: for n being Nat for r being real number for f1 being real-valued b1 -element FinSequence holds mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) let n be Nat; ::_thesis: for r being real number for f1 being real-valued n -element FinSequence holds mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) let r be real number ; ::_thesis: for f1 being real-valued n -element FinSequence holds mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) let f1 be real-valued n -element FinSequence; ::_thesis: mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) set p = (0.REAL n) +* (x,r); A1: dom f1 = Seg n by FINSEQ_1:89; A2: dom ((0.REAL n) +* (x,r)) = Seg n by FINSEQ_1:89; A3: dom (mlt (f1,((0.REAL n) +* (x,r)))) = (dom f1) /\ (dom ((0.REAL n) +* (x,r))) by VALUED_1:def_4; A4: dom ((0.REAL n) +* (x,((f1 . x) * r))) = dom (0.REAL n) by FUNCT_7:30; A5: dom (0.REAL n) = Seg n by FINSEQ_1:89; now__::_thesis:_for_z_being_set_st_z_in_dom_(mlt_(f1,((0.REAL_n)_+*_(x,r))))_holds_ (mlt_(f1,((0.REAL_n)_+*_(x,r))))_._z_=_((0.REAL_n)_+*_(x,((f1_._x)_*_r)))_._z let z be set ; ::_thesis: ( z in dom (mlt (f1,((0.REAL n) +* (x,r)))) implies (mlt (f1,((0.REAL n) +* (x,r)))) . b1 = ((0.REAL n) +* (x,((f1 . x) * r))) . b1 ) assume A6: z in dom (mlt (f1,((0.REAL n) +* (x,r)))) ; ::_thesis: (mlt (f1,((0.REAL n) +* (x,r)))) . b1 = ((0.REAL n) +* (x,((f1 . x) * r))) . b1 A7: (mlt (f1,((0.REAL n) +* (x,r)))) . z = (f1 . z) * (((0.REAL n) +* (x,r)) . z) by VALUED_1:5; percases ( z = x or z <> x ) ; supposeA8: z = x ; ::_thesis: (mlt (f1,((0.REAL n) +* (x,r)))) . b1 = ((0.REAL n) +* (x,((f1 . x) * r))) . b1 hence (mlt (f1,((0.REAL n) +* (x,r)))) . z = (f1 . z) * r by A1, A2, A3, A5, A6, A7, FUNCT_7:31 .= ((0.REAL n) +* (x,((f1 . x) * r))) . z by A1, A2, A3, A5, A6, A8, FUNCT_7:31 ; ::_thesis: verum end; supposeA9: z <> x ; ::_thesis: (mlt (f1,((0.REAL n) +* (x,r)))) . b1 = ((0.REAL n) +* (x,((f1 . x) * r))) . b1 hence (mlt (f1,((0.REAL n) +* (x,r)))) . z = (f1 . z) * ((0.REAL n) . z) by A7, FUNCT_7:32 .= (f1 . z) * ((n |-> 0) . z) .= ((0.REAL n) +* (x,((f1 . x) * r))) . z by A9, FUNCT_7:32 ; ::_thesis: verum end; end; end; hence mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) by A4, A5, FINSEQ_1:89, FUNCT_1:2; ::_thesis: verum end; theorem :: TOPREALC:16 for x being set for n being Nat for r being real number for f1 being real-valued b2 -element FinSequence holds |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r proof let x be set ; ::_thesis: for n being Nat for r being real number for f1 being real-valued b1 -element FinSequence holds |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r let n be Nat; ::_thesis: for r being real number for f1 being real-valued n -element FinSequence holds |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r let r be real number ; ::_thesis: for f1 being real-valued n -element FinSequence holds |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r let f1 be real-valued n -element FinSequence; ::_thesis: |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r A1: mlt (f1,((0.REAL n) +* (x,r))) = (0.REAL n) +* (x,((f1 . x) * r)) by Th15; A2: dom f1 = Seg n by FINSEQ_1:89; A3: n in NAT by ORDINAL1:def_12; A4: (f1 . x) * r in REAL by XREAL_0:def_1; percases ( x in dom f1 or not x in dom f1 ) ; suppose x in dom f1 ; ::_thesis: |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r hence |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r by A1, A2, A3, A4, JORDAN2B:10; ::_thesis: verum end; suppose not x in dom f1 ; ::_thesis: |(f1,((0.REAL n) +* (x,r)))| = (f1 . x) * r then A5: f1 . x = 0 by FUNCT_1:def_2; hence |(f1,((0.REAL n) +* (x,r)))| = Sum (0.REAL n) by A1, Th14 .= (f1 . x) * r by A5, A3, JORDAN2B:9 ; ::_thesis: verum end; end; end; theorem Th17: :: TOPREALC:17 for i, n being Nat for c being complex number for g1 being complex-valued b2 -element FinSequence holds (g1 +* (i,c)) - g1 = (0* n) +* (i,(c - (g1 . i))) proof let i, n be Nat; ::_thesis: for c being complex number for g1 being complex-valued n -element FinSequence holds (g1 +* (i,c)) - g1 = (0* n) +* (i,(c - (g1 . i))) let c be complex number ; ::_thesis: for g1 being complex-valued n -element FinSequence holds (g1 +* (i,c)) - g1 = (0* n) +* (i,(c - (g1 . i))) let g1 be complex-valued n -element FinSequence; ::_thesis: (g1 +* (i,c)) - g1 = (0* n) +* (i,(c - (g1 . i))) A1: dom ((g1 +* (i,c)) - g1) = (dom (g1 +* (i,c))) /\ (dom g1) by VALUED_1:12; A2: dom (0* n) = Seg n by FINSEQ_1:89; A3: dom g1 = Seg n by FINSEQ_1:89; A4: dom (g1 +* (i,c)) = dom g1 by FUNCT_7:30; hence dom ((g1 +* (i,c)) - g1) = dom ((0* n) +* (i,(c - (g1 . i)))) by A1, A3, FINSEQ_1:89; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((g1 +* (i,c)) - g1) or ((g1 +* (i,c)) - g1) . b1 = ((0* n) +* (i,(c - (g1 . i)))) . b1 ) let x be set ; ::_thesis: ( not x in dom ((g1 +* (i,c)) - g1) or ((g1 +* (i,c)) - g1) . x = ((0* n) +* (i,(c - (g1 . i)))) . x ) assume A5: x in dom ((g1 +* (i,c)) - g1) ; ::_thesis: ((g1 +* (i,c)) - g1) . x = ((0* n) +* (i,(c - (g1 . i)))) . x then A6: ((g1 +* (i,c)) - g1) . x = ((g1 +* (i,c)) . x) - (g1 . x) by VALUED_1:13; percases ( x = i or x <> i ) ; supposeA7: x = i ; ::_thesis: ((g1 +* (i,c)) - g1) . x = ((0* n) +* (i,(c - (g1 . i)))) . x hence ((g1 +* (i,c)) - g1) . x = c - (g1 . x) by A6, A1, A5, A4, FUNCT_7:31 .= ((0* n) +* (i,(c - (g1 . i)))) . x by A1, A2, A3, A5, A4, A7, FUNCT_7:31 ; ::_thesis: verum end; supposeA8: x <> i ; ::_thesis: ((g1 +* (i,c)) - g1) . x = ((0* n) +* (i,(c - (g1 . i)))) . x hence ((g1 +* (i,c)) - g1) . x = (g1 . x) - (g1 . x) by A6, FUNCT_7:32 .= (n |-> 0) . x .= ((0* n) +* (i,(c - (g1 . i)))) . x by A8, FUNCT_7:32 ; ::_thesis: verum end; end; end; theorem :: TOPREALC:18 for r being real number holds |.<*r*>.| = abs r proof let r be real number ; ::_thesis: |.<*r*>.| = abs r set f = <*r*>; sqr <*r*> = <*(r ^2)*> by RVSUM_1:55; then Sum (sqr <*r*>) = r ^2 by RVSUM_1:73; hence |.<*r*>.| = abs r by COMPLEX1:72; ::_thesis: verum end; theorem :: TOPREALC:19 for f being real-valued FinSequence holds f is FinSequence of REAL by RVSUM_1:145; theorem :: TOPREALC:20 for f being real-valued FinSequence st |.f.| <> 0 holds ex i being Nat st ( i in dom f & f . i <> 0 ) proof let f be real-valued FinSequence; ::_thesis: ( |.f.| <> 0 implies ex i being Nat st ( i in dom f & f . i <> 0 ) ) assume |.f.| <> 0 ; ::_thesis: ex i being Nat st ( i in dom f & f . i <> 0 ) then consider i being Element of NAT such that A1: i in dom (sqr f) and A2: (sqr f) . i <> 0 by PRVECT_2:3, SQUARE_1:17; take i ; ::_thesis: ( i in dom f & f . i <> 0 ) thus i in dom f by A1, VALUED_1:11; ::_thesis: f . i <> 0 (sqr f) . i = (f . i) ^2 by VALUED_1:11; hence f . i <> 0 by A2; ::_thesis: verum end; theorem Th21: :: TOPREALC:21 for f being real-valued FinSequence holds abs (Sum f) <= Sum (abs f) proof let f be real-valued FinSequence; ::_thesis: abs (Sum f) <= Sum (abs f) defpred S1[ set ] means ex F being real-valued FinSequence st ( F = \$1 & abs (Sum F) <= Sum (abs F) ); A1: S1[ <*> REAL] proof take <*> REAL ; ::_thesis: ( <*> REAL = <*> REAL & abs (Sum (<*> REAL)) <= Sum (abs (<*> REAL)) ) thus ( <*> REAL = <*> REAL & abs (Sum (<*> REAL)) <= Sum (abs (<*> REAL)) ) by ABSVALUE:2, RVSUM_1:72; ::_thesis: verum end; A2: for p being FinSequence of REAL for x being Element of REAL st S1[p] holds S1[p ^ <*x*>] proof let p be FinSequence of REAL ; ::_thesis: for x being Element of REAL st S1[p] holds S1[p ^ <*x*>] let x be Element of REAL ; ::_thesis: ( S1[p] implies S1[p ^ <*x*>] ) given F being real-valued FinSequence such that A3: F = p and A4: abs (Sum F) <= Sum (abs F) ; ::_thesis: S1[p ^ <*x*>] A5: (abs (Sum F)) + (abs x) <= (Sum (abs F)) + (abs x) by A4, XREAL_1:6; take G = p ^ <*x*>; ::_thesis: ( G = p ^ <*x*> & abs (Sum G) <= Sum (abs G) ) A6: abs <*x*> = <*(abs x)*> by JORDAN2B:19; A7: Sum <*(abs x)*> = abs x by RVSUM_1:73; abs G = (abs p) ^ (abs <*x*>) by TOPREAL7:11; then A8: Sum (abs G) = (Sum (abs p)) + (abs x) by A6, A7, RVSUM_1:75; Sum G = (Sum p) + x by RVSUM_1:74; then abs (Sum G) <= (abs (Sum p)) + (abs x) by COMPLEX1:56; hence ( G = p ^ <*x*> & abs (Sum G) <= Sum (abs G) ) by A8, A3, A5, XXREAL_0:2; ::_thesis: verum end; A9: for p being FinSequence of REAL holds S1[p] from FINSEQ_2:sch_2(A1, A2); f is FinSequence of REAL by RVSUM_1:145; then S1[f] by A9; hence abs (Sum f) <= Sum (abs f) ; ::_thesis: verum end; Lm1: for n being Nat for f being real-valued b1 -element FinSequence holds f is Element of n -tuples_on REAL proof let n be Nat; ::_thesis: for f being real-valued n -element FinSequence holds f is Element of n -tuples_on REAL let f be real-valued n -element FinSequence; ::_thesis: f is Element of n -tuples_on REAL A1: f is FinSequence of REAL by RVSUM_1:145; len f = n by CARD_1:def_7; hence f is Element of n -tuples_on REAL by A1, FINSEQ_2:92; ::_thesis: verum end; theorem :: TOPREALC:22 for A being non empty 1-sorted for B being 1 -element 1-sorted for t being Point of B for f being Function of A,B holds f = A --> t proof let A be non empty 1-sorted ; ::_thesis: for B being 1 -element 1-sorted for t being Point of B for f being Function of A,B holds f = A --> t let B be 1 -element 1-sorted ; ::_thesis: for t being Point of B for f being Function of A,B holds f = A --> t let t be Point of B; ::_thesis: for f being Function of A,B holds f = A --> t let f be Function of A,B; ::_thesis: f = A --> t let a be Element of A; :: according to FUNCT_2:def_8 ::_thesis: f . a = (A --> t) . a thus f . a = (A --> t) . a by STRUCT_0:def_10; ::_thesis: verum end; registration let n be non zero Nat; let i be Element of Seg n; let T be non empty real-membered TopSpace; cluster proj (((Seg n) --> T),i) -> real-valued ; coherence proj (((Seg n) --> T),i) is real-valued proof ((Seg n) --> T) . i = T by FUNCOP_1:7; hence proj (((Seg n) --> T),i) is real-valued ; ::_thesis: verum end; end; definition let n be Nat; let p be Element of REAL n; let r be real number ; :: original: (/) redefine funcp (/) r -> Element of REAL n; coherence p (/) r is Element of REAL n proof A1: p (/) r is FinSequence of REAL by RVSUM_1:145; A2: len p = n by CARD_1:def_7; Seg (len (p (/) r)) = dom (p (/) r) by FINSEQ_1:def_3 .= dom p by VALUED_1:def_5 .= Seg n by A2, FINSEQ_1:def_3 ; then len (p (/) r) = n by FINSEQ_1:6; hence p (/) r is Element of REAL n by A1, FINSEQ_2:92; ::_thesis: verum end; end; theorem Th23: :: TOPREALC:23 for m being Nat for r being real number for p, q being Point of (TOP-REAL m) holds ( p in Ball (q,r) iff - p in Ball ((- q),r) ) proof let m be Nat; ::_thesis: for r being real number for p, q being Point of (TOP-REAL m) holds ( p in Ball (q,r) iff - p in Ball ((- q),r) ) let r be real number ; ::_thesis: for p, q being Point of (TOP-REAL m) holds ( p in Ball (q,r) iff - p in Ball ((- q),r) ) let p, q be Point of (TOP-REAL m); ::_thesis: ( p in Ball (q,r) iff - p in Ball ((- q),r) ) A1: m in NAT by ORDINAL1:def_12; A2: now__::_thesis:_for_a,_b_being_Point_of_(TOP-REAL_m)_st_a_in_Ball_(b,r)_holds_ -_a_in_Ball_((-_b),r) let a, b be Point of (TOP-REAL m); ::_thesis: ( a in Ball (b,r) implies - a in Ball ((- b),r) ) assume a in Ball (b,r) ; ::_thesis: - a in Ball ((- b),r) then A3: |.(a - b).| < r by A1, TOPREAL9:7; (- a) - (- b) = (- a) + (- (- b)) .= - (a - b) by RLVECT_1:31 ; then |.((- a) - (- b)).| = |.(a - b).| by EUCLID:71; hence - a in Ball ((- b),r) by A1, A3, TOPREAL9:7; ::_thesis: verum end; ( - (- p) = p & - (- q) = q ) ; hence ( p in Ball (q,r) iff - p in Ball ((- q),r) ) by A2; ::_thesis: verum end; definition let S be 1-sorted ; attrS is complex-functions-membered means :Def1: :: TOPREALC:def 1 the carrier of S is complex-functions-membered ; attrS is real-functions-membered means :Def2: :: TOPREALC:def 2 the carrier of S is real-functions-membered ; end; :: deftheorem Def1 defines complex-functions-membered TOPREALC:def_1_:_ for S being 1-sorted holds ( S is complex-functions-membered iff the carrier of S is complex-functions-membered ); :: deftheorem Def2 defines real-functions-membered TOPREALC:def_2_:_ for S being 1-sorted holds ( S is real-functions-membered iff the carrier of S is real-functions-membered ); registration let n be Nat; cluster TOP-REAL n -> real-functions-membered ; coherence TOP-REAL n is real-functions-membered proof let x be set ; :: according to VALUED_2:def_4,TOPREALC:def_2 ::_thesis: ( not x in the carrier of (TOP-REAL n) or x is set ) assume x in the carrier of (TOP-REAL n) ; ::_thesis: x is set then reconsider x = x as Point of (TOP-REAL n) ; x is real-valued ; hence x is set ; ::_thesis: verum end; end; registration cluster TOP-REAL 0 -> real-membered ; coherence TOP-REAL 0 is real-membered proof thus the carrier of (TOP-REAL 0) is real-membered by EUCLID:22, EUCLID:77; :: according to TOPMETR:def_8 ::_thesis: verum end; end; registration cluster TOP-REAL 0 -> trivial ; coherence TOP-REAL 0 is trivial by EUCLID:22, EUCLID:77; end; registration cluster real-functions-membered -> complex-functions-membered for 1-sorted ; coherence for b1 being 1-sorted st b1 is real-functions-membered holds b1 is complex-functions-membered proof let S be 1-sorted ; ::_thesis: ( S is real-functions-membered implies S is complex-functions-membered ) assume A1: the carrier of S is real-functions-membered ; :: according to TOPREALC:def_2 ::_thesis: S is complex-functions-membered let x be set ; :: according to VALUED_2:def_2,TOPREALC:def_1 ::_thesis: ( not x in the carrier of S or x is set ) thus ( not x in the carrier of S or x is set ) by A1; ::_thesis: verum end; end; registration cluster strict non empty real-functions-membered for 1-sorted ; existence ex b1 being 1-sorted st ( b1 is strict & not b1 is empty & b1 is real-functions-membered ) proof take S = 1-sorted(# { the real-valued Function} #); ::_thesis: ( S is strict & not S is empty & S is real-functions-membered ) thus S is strict ; ::_thesis: ( not S is empty & S is real-functions-membered ) thus not the carrier of S is empty ; :: according to STRUCT_0:def_1 ::_thesis: S is real-functions-membered let x be set ; :: according to VALUED_2:def_4,TOPREALC:def_2 ::_thesis: ( not x in the carrier of S or x is set ) thus ( not x in the carrier of S or x is set ) ; ::_thesis: verum end; end; registration let S be complex-functions-membered 1-sorted ; cluster the carrier of S -> complex-functions-membered ; coherence the carrier of S is complex-functions-membered by Def1; end; registration let S be real-functions-membered 1-sorted ; cluster the carrier of S -> real-functions-membered ; coherence the carrier of S is real-functions-membered by Def2; end; registration cluster non empty strict TopSpace-like real-functions-membered for TopStruct ; existence ex b1 being TopSpace st ( b1 is strict & not b1 is empty & b1 is real-functions-membered ) proof take TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) ; ::_thesis: ( TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is strict & not TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is empty & TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is real-functions-membered ) thus ( TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is strict & not TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is empty & TopStruct(# the carrier of (TOP-REAL 1), the topology of (TOP-REAL 1) #) is real-functions-membered ) by Def2; ::_thesis: verum end; end; registration let S be complex-functions-membered TopSpace; cluster -> complex-functions-membered for SubSpace of S; coherence for b1 being SubSpace of S holds b1 is complex-functions-membered proof let A be SubSpace of S; ::_thesis: A is complex-functions-membered let x be set ; :: according to VALUED_2:def_2,TOPREALC:def_1 ::_thesis: ( not x in the carrier of A or x is set ) the carrier of A c= the carrier of S by BORSUK_1:1; hence ( not x in the carrier of A or x is set ) ; ::_thesis: verum end; end; registration let S be real-functions-membered TopSpace; cluster -> real-functions-membered for SubSpace of S; coherence for b1 being SubSpace of S holds b1 is real-functions-membered proof let A be SubSpace of S; ::_thesis: A is real-functions-membered let x be set ; :: according to VALUED_2:def_4,TOPREALC:def_2 ::_thesis: ( not x in the carrier of A or x is set ) the carrier of A c= the carrier of S by BORSUK_1:1; hence ( not x in the carrier of A or x is set ) ; ::_thesis: verum end; end; definition let X be complex-functions-membered set ; func (-) X -> complex-functions-membered set means :Def3: :: TOPREALC:def 3 for f being complex-valued Function holds ( - f in it iff f in X ); existence ex b1 being complex-functions-membered set st for f being complex-valued Function holds ( - f in b1 iff f in X ) proof set F = { (- f) where f is Element of X : f in X } ; { (- f) where f is Element of X : f in X } is complex-functions-membered proof let x be set ; :: according to VALUED_2:def_2 ::_thesis: ( not x in { (- f) where f is Element of X : f in X } or x is set ) assume x in { (- f) where f is Element of X : f in X } ; ::_thesis: x is set then ex f being Element of X st ( x = - f & f in X ) ; hence x is set ; ::_thesis: verum end; then reconsider F = { (- f) where f is Element of X : f in X } as complex-functions-membered set ; take F ; ::_thesis: for f being complex-valued Function holds ( - f in F iff f in X ) let f be complex-valued Function; ::_thesis: ( - f in F iff f in X ) hereby ::_thesis: ( f in X implies - f in F ) assume - f in F ; ::_thesis: f in X then ex g being Element of X st ( - f = - g & g in X ) ; hence f in X by RVSUM_1:24; ::_thesis: verum end; thus ( f in X implies - f in F ) ; ::_thesis: verum end; uniqueness for b1, b2 being complex-functions-membered set st ( for f being complex-valued Function holds ( - f in b1 iff f in X ) ) & ( for f being complex-valued Function holds ( - f in b2 iff f in X ) ) holds b1 = b2 proof let A, B be complex-functions-membered set ; ::_thesis: ( ( for f being complex-valued Function holds ( - f in A iff f in X ) ) & ( for f being complex-valued Function holds ( - f in B iff f in X ) ) implies A = B ) assume that A1: for f being complex-valued Function holds ( - f in A iff f in X ) and A2: for f being complex-valued Function holds ( - f in B iff f in X ) ; ::_thesis: A = B thus A c= B :: according to XBOOLE_0:def_10 ::_thesis: B c= A proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in B ) assume A3: x in A ; ::_thesis: x in B then reconsider x = x as Element of A ; A4: - (- x) = x ; then - x in X by A1, A3; hence x in B by A2, A4; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in B or x in A ) assume A5: x in B ; ::_thesis: x in A then reconsider x = x as Element of B ; A6: - (- x) = x ; then - x in X by A2, A5; hence x in A by A1, A6; ::_thesis: verum end; involutiveness for b1, b2 being complex-functions-membered set st ( for f being complex-valued Function holds ( - f in b1 iff f in b2 ) ) holds for f being complex-valued Function holds ( - f in b2 iff f in b1 ) proof let r, A be complex-functions-membered set ; ::_thesis: ( ( for f being complex-valued Function holds ( - f in r iff f in A ) ) implies for f being complex-valued Function holds ( - f in A iff f in r ) ) assume A7: for f being complex-valued Function holds ( - f in r iff f in A ) ; ::_thesis: for f being complex-valued Function holds ( - f in A iff f in r ) let f be complex-valued Function; ::_thesis: ( - f in A iff f in r ) hereby ::_thesis: ( f in r implies - f in A ) assume - f in A ; ::_thesis: f in r then - (- f) in r by A7; hence f in r ; ::_thesis: verum end; assume f in r ; ::_thesis: - f in A then - (- f) in r ; hence - f in A by A7; ::_thesis: verum end; end; :: deftheorem Def3 defines (-) TOPREALC:def_3_:_ for X, b2 being complex-functions-membered set holds ( b2 = (-) X iff for f being complex-valued Function holds ( - f in b2 iff f in X ) ); registration let X be empty set ; cluster (-) X -> empty complex-functions-membered ; coherence (-) X is empty proof assume not (-) X is empty ; ::_thesis: contradiction then consider x being set such that A1: x in (-) X by XBOOLE_0:def_1; reconsider x = x as Element of (-) X by A1; - (- x) = x ; hence contradiction by A1, Def3; ::_thesis: verum end; end; registration let X be non empty complex-functions-membered set ; cluster (-) X -> non empty complex-functions-membered ; coherence not (-) X is empty proof consider x being set such that A1: x in X by XBOOLE_0:def_1; reconsider x = x as Element of X by A1; - x in (-) X by Def3; hence not (-) X is empty ; ::_thesis: verum end; end; theorem Th24: :: TOPREALC:24 for X being complex-functions-membered set for f being complex-valued Function holds ( - f in X iff f in (-) X ) proof let X be complex-functions-membered set ; ::_thesis: for f being complex-valued Function holds ( - f in X iff f in (-) X ) let f be complex-valued Function; ::_thesis: ( - f in X iff f in (-) X ) ( - (- f) = f & (-) ((-) X) = X ) ; hence ( - f in X iff f in (-) X ) by Def3; ::_thesis: verum end; registration let X be real-functions-membered set ; cluster (-) X -> complex-functions-membered real-functions-membered ; coherence (-) X is real-functions-membered proof let x be set ; :: according to VALUED_2:def_4 ::_thesis: ( not x in (-) X or x is set ) assume A1: x in (-) X ; ::_thesis: x is set then reconsider x = x as Element of (-) X ; A2: - x in X by A1, Th24; ( - (- x) = x & (-) ((-) X) = X ) ; hence x is set by A2; ::_thesis: verum end; end; theorem Th25: :: TOPREALC:25 for n being Nat for X being Subset of (TOP-REAL n) holds - X = (-) X proof let n be Nat; ::_thesis: for X being Subset of (TOP-REAL n) holds - X = (-) X set T = TOP-REAL n; let X be Subset of (TOP-REAL n); ::_thesis: - X = (-) X for f being complex-valued Function holds ( - f in - X iff f in X ) proof let f be complex-valued Function; ::_thesis: ( - f in - X iff f in X ) hereby ::_thesis: ( f in X implies - f in - X ) assume - f in - X ; ::_thesis: f in X then consider v being Element of (TOP-REAL n) such that A1: - f = (- 1) * v and A2: v in X ; reconsider g = - f as Element of (TOP-REAL n) by A1; - g = - (- v) by A1 .= v ; hence f in X by A2; ::_thesis: verum end; assume A3: f in X ; ::_thesis: - f in - X then reconsider g = f as Element of (TOP-REAL n) ; - g = (- 1) * g ; hence - f in - X by A3; ::_thesis: verum end; hence - X = (-) X by Def3; ::_thesis: verum end; definition let n be Nat; let X be Subset of (TOP-REAL n); :: original: (-) redefine func (-) X -> Subset of (TOP-REAL n); coherence (-) X is Subset of (TOP-REAL n) proof - X = (-) X by Th25; hence (-) X is Subset of (TOP-REAL n) ; ::_thesis: verum end; end; registration let n be Nat; let X be open Subset of (TOP-REAL n); cluster (-) X -> open for Subset of (TOP-REAL n); coherence for b1 being Subset of (TOP-REAL n) st b1 = (-) X holds b1 is open proof reconsider n = n as Element of NAT by ORDINAL1:def_12; reconsider Y = (-) X as Subset of (TOP-REAL n) ; for p being Point of (TOP-REAL n) st p in Y holds ex r being positive real number st Ball (p,r) c= Y proof let p be Point of (TOP-REAL n); ::_thesis: ( p in Y implies ex r being positive real number st Ball (p,r) c= Y ) assume p in Y ; ::_thesis: ex r being positive real number st Ball (p,r) c= Y then - p in (-) Y by Def3; then consider r being positive real number such that A1: Ball ((- p),r) c= X by TOPS_4:2; take r ; ::_thesis: Ball (p,r) c= Y let x be Element of (TOP-REAL n); :: according to LATTICE7:def_1 ::_thesis: ( not x in Ball (p,r) or x in Y ) assume x in Ball (p,r) ; ::_thesis: x in Y then - x in Ball ((- p),r) by Th23; hence x in Y by A1, Th24; ::_thesis: verum end; hence for b1 being Subset of (TOP-REAL n) st b1 = (-) X holds b1 is open by TOPS_4:2; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let x be set ; :: original: . redefine funcp . x -> Element of REAL ; coherence p . x is Element of REAL by XREAL_0:def_1; end; definition let R, S, T be non empty TopSpace; let f be Function of [:R,S:],T; let x be Point of [:R,S:]; :: original: . redefine funcf . x -> Point of T; coherence f . x is Point of T proof consider x1 being Point of R, x2 being Point of S such that A1: x = [x1,x2] by BORSUK_1:10; f . [x1,x2] is Point of T ; hence f . x is Point of T by A1; ::_thesis: verum end; end; definition let R, S, T be non empty TopSpace; let f be Function of [:R,S:],T; let r be Point of R; let s be Point of S; :: original: . redefine funcf . (r,s) -> Point of T; coherence f . (r,s) is Point of T proof f . (r,s) = f . [r,s] ; hence f . (r,s) is Point of T ; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let r be real number ; :: original: + redefine funcp + r -> Point of (TOP-REAL n); coherence r + p is Point of (TOP-REAL n) proof A1: p + r is FinSequence of REAL by RVSUM_1:145; A2: len p = n by CARD_1:def_7; Seg (len (p + r)) = dom (p + r) by FINSEQ_1:def_3 .= dom p by VALUED_1:def_2 .= Seg n by A2, FINSEQ_1:def_3 ; then len (p + r) = n by FINSEQ_1:6; then p + r is Element of REAL n by A1, FINSEQ_2:92; hence r + p is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let r be real number ; :: original: - redefine funcp - r -> Point of (TOP-REAL n); coherence p - r is Point of (TOP-REAL n) proof p + (- r) is Point of (TOP-REAL n) ; hence p - r is Point of (TOP-REAL n) ; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let r be real number ; :: original: (#) redefine funcp (#) r -> Point of (TOP-REAL n); coherence r (#) p is Point of (TOP-REAL n) proof A1: p (#) r is FinSequence of REAL by RVSUM_1:145; A2: len p = n by CARD_1:def_7; Seg (len (p (#) r)) = dom (p (#) r) by FINSEQ_1:def_3 .= dom p by VALUED_1:def_5 .= Seg n by A2, FINSEQ_1:def_3 ; then len (p (#) r) = n by FINSEQ_1:6; then p (#) r is Element of REAL n by A1, FINSEQ_2:92; hence r (#) p is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let r be real number ; :: original: (/) redefine funcp (/) r -> Point of (TOP-REAL n); coherence p (/) r is Point of (TOP-REAL n) proof reconsider f = p as Element of REAL n by EUCLID:22; f (/) r is Element of REAL n ; hence p (/) r is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; end; definition let n be Nat; let p1, p2 be Point of (TOP-REAL n); :: original: (#) redefine funcp1 (#) p2 -> Point of (TOP-REAL n); coherence p1 (#) p2 is Point of (TOP-REAL n) proof A1: p1 (#) p2 is FinSequence of REAL by RVSUM_1:145; A2: ( len p1 = n & len p2 = n ) by CARD_1:def_7; Seg (len (p1 (#) p2)) = dom (p1 (#) p2) by FINSEQ_1:def_3 .= (dom p1) /\ (dom p2) by VALUED_1:def_4 .= (Seg n) /\ (dom p2) by A2, FINSEQ_1:def_3 .= (Seg n) /\ (Seg n) by A2, FINSEQ_1:def_3 ; then len (p1 (#) p2) = n by FINSEQ_1:6; then p1 (#) p2 is Element of REAL n by A1, FINSEQ_2:92; hence p1 (#) p2 is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; commutativity for p1, p2 being Point of (TOP-REAL n) holds p1 (#) p2 = p2 (#) p1 ; end; definition let n be Nat; let p be Point of (TOP-REAL n); :: original: ^2 redefine func sqr p -> Point of (TOP-REAL n); coherence p ^2 is Point of (TOP-REAL n) proof sqr p = p (#) p ; hence p ^2 is Point of (TOP-REAL n) ; ::_thesis: verum end; end; definition let n be Nat; let p1, p2 be Point of (TOP-REAL n); :: original: /" redefine funcp1 /" p2 -> Point of (TOP-REAL n); coherence p1 /" p2 is Point of (TOP-REAL n) proof A1: p1 /" p2 is FinSequence of REAL by RVSUM_1:145; A2: ( len p1 = n & len p2 = n ) by CARD_1:def_7; Seg (len (p1 /" p2)) = dom (p1 /" p2) by FINSEQ_1:def_3 .= (dom p1) /\ (dom p2) by VALUED_1:16 .= (Seg n) /\ (dom p2) by A2, FINSEQ_1:def_3 .= (Seg n) /\ (Seg n) by A2, FINSEQ_1:def_3 ; then len (p1 /" p2) = n by FINSEQ_1:6; then p1 /" p2 is Element of REAL n by A1, FINSEQ_2:92; hence p1 /" p2 is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; end; definition let n be Nat; let p be Element of (TOP-REAL n); let x be set ; let r be real number ; :: original: +* redefine funcp +* (x,r) -> Point of (TOP-REAL n); coherence p +* (x,r) is Point of (TOP-REAL n) proof A1: p +* (x,r) is FinSequence of REAL by RVSUM_1:145; A2: len p = n by CARD_1:def_7; Seg (len (p +* (x,r))) = dom (p +* (x,r)) by FINSEQ_1:def_3 .= dom p by FUNCT_7:30 .= Seg n by A2, FINSEQ_1:def_3 ; then len (p +* (x,r)) = n by FINSEQ_1:6; then p +* (x,r) is Element of REAL n by A1, FINSEQ_2:92; hence p +* (x,r) is Point of (TOP-REAL n) by EUCLID:22; ::_thesis: verum end; end; theorem :: TOPREALC:26 for n being Nat for r being real number for a, o being Point of (TOP-REAL n) st n <> 0 & a in Ball (o,r) holds abs (Sum (a - o)) < n * r proof let n be Nat; ::_thesis: for r being real number for a, o being Point of (TOP-REAL n) st n <> 0 & a in Ball (o,r) holds abs (Sum (a - o)) < n * r let r be real number ; ::_thesis: for a, o being Point of (TOP-REAL n) st n <> 0 & a in Ball (o,r) holds abs (Sum (a - o)) < n * r let a, o be Point of (TOP-REAL n); ::_thesis: ( n <> 0 & a in Ball (o,r) implies abs (Sum (a - o)) < n * r ) set R1 = a - o; set R2 = n |-> r; assume that A1: n <> 0 and A2: a in Ball (o,r) ; ::_thesis: abs (Sum (a - o)) < n * r A3: Sum (n |-> r) = n * r by RVSUM_1:80; A4: ( abs (a - o) is Element of n -tuples_on REAL & n |-> r is Element of n -tuples_on REAL ) by Lm1; A5: for j being Nat st j in Seg n holds (abs (a - o)) . j < (n |-> r) . j proof let j be Nat; ::_thesis: ( j in Seg n implies (abs (a - o)) . j < (n |-> r) . j ) assume j in Seg n ; ::_thesis: (abs (a - o)) . j < (n |-> r) . j then A6: (n |-> r) . j = r by FINSEQ_2:57; abs ((a - o) . j) < r by A2, EUCLID_9:9; hence (abs (a - o)) . j < (n |-> r) . j by A6, VALUED_1:18; ::_thesis: verum end; then A7: for j being Nat st j in Seg n holds (abs (a - o)) . j <= (n |-> r) . j ; ex j being Nat st ( j in Seg n & (abs (a - o)) . j < (n |-> r) . j ) proof take 1 ; ::_thesis: ( 1 in Seg n & (abs (a - o)) . 1 < (n |-> r) . 1 ) 1 <= n by A1, NAT_1:14; hence 1 in Seg n ; ::_thesis: (abs (a - o)) . 1 < (n |-> r) . 1 hence (abs (a - o)) . 1 < (n |-> r) . 1 by A5; ::_thesis: verum end; then A8: Sum (abs (a - o)) < Sum (n |-> r) by A7, A4, RVSUM_1:83; abs (Sum (a - o)) <= Sum (abs (a - o)) by Th21; hence abs (Sum (a - o)) < n * r by A3, A8, XXREAL_0:2; ::_thesis: verum end; registration let n be Nat; cluster Euclid n -> real-functions-membered ; coherence Euclid n is real-functions-membered proof let x be set ; :: according to VALUED_2:def_4,TOPREALC:def_2 ::_thesis: ( not x in the carrier of (Euclid n) or x is set ) assume x in the carrier of (Euclid n) ; ::_thesis: x is set then reconsider x = x as Point of (Euclid n) ; x is real-valued ; hence x is set ; ::_thesis: verum end; end; theorem :: TOPREALC:27 for V being non empty right_complementable add-associative right_zeroed addLoopStr for v, u being Element of V holds (v + u) - u = v proof let V be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for v, u being Element of V holds (v + u) - u = v let v, u be Element of V; ::_thesis: (v + u) - u = v thus (v + u) - u = v + (u - u) by RLVECT_1:28 .= v + (0. V) by RLVECT_1:5 .= v by RLVECT_1:4 ; ::_thesis: verum end; theorem :: TOPREALC:28 for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr for v, u being Element of V holds (v - u) + u = v proof let V be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for v, u being Element of V holds (v - u) + u = v let v, u be Element of V; ::_thesis: (v - u) + u = v thus (v - u) + u = v - (u - u) by RLVECT_1:29 .= v - (0. V) by RLVECT_1:5 .= v by RLVECT_1:13 ; ::_thesis: verum end; theorem Th29: :: TOPREALC:29 for X being set for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [+] c = f <+> ((dom f) --> c) proof let X be set ; ::_thesis: for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [+] c = f <+> ((dom f) --> c) let c be complex number ; ::_thesis: for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [+] c = f <+> ((dom f) --> c) let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y holds f [+] c = f <+> ((dom f) --> c) let f be PartFunc of X,Y; ::_thesis: f [+] c = f <+> ((dom f) --> c) set g = (dom f) --> c; A1: dom (f [+] c) = dom f by VALUED_2:def_37; A2: dom (f <+> ((dom f) --> c)) = (dom f) /\ (dom ((dom f) --> c)) by VALUED_2:def_41; A3: dom ((dom f) --> c) = dom f by FUNCOP_1:13; now__::_thesis:_for_x_being_set_st_x_in_dom_(f_[+]_c)_holds_ (f_[+]_c)_._x_=_(f_<+>_((dom_f)_-->_c))_._x let x be set ; ::_thesis: ( x in dom (f [+] c) implies (f [+] c) . x = (f <+> ((dom f) --> c)) . x ) assume A4: x in dom (f [+] c) ; ::_thesis: (f [+] c) . x = (f <+> ((dom f) --> c)) . x hence (f [+] c) . x = (f . x) + c by VALUED_2:def_37 .= (f . x) + (((dom f) --> c) . x) by A1, A4, FUNCOP_1:7 .= (f <+> ((dom f) --> c)) . x by A1, A2, A3, A4, VALUED_2:def_41 ; ::_thesis: verum end; hence f [+] c = f <+> ((dom f) --> c) by A1, A2, A3, FUNCT_1:2; ::_thesis: verum end; theorem :: TOPREALC:30 for X being set for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [-] c = f <-> ((dom f) --> c) proof let X be set ; ::_thesis: for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [-] c = f <-> ((dom f) --> c) let c be complex number ; ::_thesis: for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [-] c = f <-> ((dom f) --> c) let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y holds f [-] c = f <-> ((dom f) --> c) let f be PartFunc of X,Y; ::_thesis: f [-] c = f <-> ((dom f) --> c) set g = (dom f) --> c; A1: dom (f [-] c) = dom f by VALUED_2:def_37; A2: dom (f <-> ((dom f) --> c)) = (dom f) /\ (dom ((dom f) --> c)) by VALUED_2:61; A3: dom ((dom f) --> c) = dom f by FUNCOP_1:13; now__::_thesis:_for_x_being_set_st_x_in_dom_(f_[-]_c)_holds_ (f_[-]_c)_._x_=_(f_<->_((dom_f)_-->_c))_._x let x be set ; ::_thesis: ( x in dom (f [-] c) implies (f [-] c) . x = (f <-> ((dom f) --> c)) . x ) assume A4: x in dom (f [-] c) ; ::_thesis: (f [-] c) . x = (f <-> ((dom f) --> c)) . x hence (f [-] c) . x = (f . x) - c by VALUED_2:def_37 .= (f . x) - (((dom f) --> c) . x) by A1, A4, FUNCOP_1:7 .= (f <-> ((dom f) --> c)) . x by A1, A2, A3, A4, VALUED_2:62 ; ::_thesis: verum end; hence f [-] c = f <-> ((dom f) --> c) by A1, A2, A3, FUNCT_1:2; ::_thesis: verum end; theorem Th31: :: TOPREALC:31 for X being set for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [#] c = f <#> ((dom f) --> c) proof let X be set ; ::_thesis: for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [#] c = f <#> ((dom f) --> c) let c be complex number ; ::_thesis: for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [#] c = f <#> ((dom f) --> c) let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y holds f [#] c = f <#> ((dom f) --> c) let f be PartFunc of X,Y; ::_thesis: f [#] c = f <#> ((dom f) --> c) set g = (dom f) --> c; A1: dom (f [#] c) = dom f by VALUED_2:def_39; A2: dom (f <#> ((dom f) --> c)) = (dom f) /\ (dom ((dom f) --> c)) by VALUED_2:def_43; A3: dom ((dom f) --> c) = dom f by FUNCOP_1:13; now__::_thesis:_for_x_being_set_st_x_in_dom_(f_[#]_c)_holds_ (f_[#]_c)_._x_=_(f_<#>_((dom_f)_-->_c))_._x let x be set ; ::_thesis: ( x in dom (f [#] c) implies (f [#] c) . x = (f <#> ((dom f) --> c)) . x ) assume A4: x in dom (f [#] c) ; ::_thesis: (f [#] c) . x = (f <#> ((dom f) --> c)) . x hence (f [#] c) . x = c (#) (f . x) by VALUED_2:def_39 .= (f . x) (#) (((dom f) --> c) . x) by A1, A4, FUNCOP_1:7 .= (f <#> ((dom f) --> c)) . x by A1, A2, A3, A4, VALUED_2:def_43 ; ::_thesis: verum end; hence f [#] c = f <#> ((dom f) --> c) by A1, A2, A3, FUNCT_1:2; ::_thesis: verum end; theorem :: TOPREALC:32 for X being set for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [/] c = f ((dom f) --> c) proof let X be set ; ::_thesis: for c being complex number for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [/] c = f ((dom f) --> c) let c be complex number ; ::_thesis: for Y being complex-functions-membered set for f being PartFunc of X,Y holds f [/] c = f ((dom f) --> c) let Y be complex-functions-membered set ; ::_thesis: for f being PartFunc of X,Y holds f [/] c = f ((dom f) --> c) let f be PartFunc of X,Y; ::_thesis: f [/] c = f ((dom f) --> c) set g = (dom f) --> c; A1: dom (f [/] c) = dom f by VALUED_2:def_39; A2: dom (f ((dom f) --> c)) = (dom f) /\ (dom ((dom f) --> c)) by VALUED_2:71; A3: dom ((dom f) --> c) = dom f by FUNCOP_1:13; now__::_thesis:_for_x_being_set_st_x_in_dom_(f_[/]_c)_holds_ (f_[/]_c)_._x_=_(f__((dom_f)_-->_c))_._x let x be set ; ::_thesis: ( x in dom (f [/] c) implies (f [/] c) . x = (f ((dom f) --> c)) . x ) assume A4: x in dom (f [/] c) ; ::_thesis: (f [/] c) . x = (f ((dom f) --> c)) . x hence (f [/] c) . x = (f . x) (/) c by VALUED_2:56 .= (f . x) (/) (((dom f) --> c) . x) by A1, A4, FUNCOP_1:7 .= (f ((dom f) --> c)) . x by A1, A2, A3, A4, VALUED_2:72 ; ::_thesis: verum end; hence f [/] c = f ((dom f) --> c) by A1, A2, A3, FUNCT_1:2; ::_thesis: verum end; registration let D be complex-functions-membered set ; let f, g be FinSequence of D; clusterf <++> g -> FinSequence-like ; coherence f <++> g is FinSequence-like proof dom (f <++> g) = (dom f) /\ (dom g) by VALUED_2:def_45; hence f <++> g is FinSequence-like by VALUED_1:19; ::_thesis: verum end; clusterf <--> g -> FinSequence-like ; coherence f <--> g is FinSequence-like proof dom (f <--> g) = (dom f) /\ (dom g) by VALUED_2:def_46; hence f <--> g is FinSequence-like by VALUED_1:19; ::_thesis: verum end; clusterf <##> g -> FinSequence-like ; coherence f <##> g is FinSequence-like proof dom (f <##> g) = (dom f) /\ (dom g) by VALUED_2:def_47; hence f <##> g is FinSequence-like by VALUED_1:19; ::_thesis: verum end; clusterf g -> FinSequence-like ; coherence f g is FinSequence-like proof dom (f g) = (dom f) /\ (dom g) by VALUED_2:def_48; hence f g is FinSequence-like by VALUED_1:19; ::_thesis: verum end; end; theorem :: TOPREALC:33 for X being set for n being Nat for f being Function of X,(TOP-REAL n) holds <-> f is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f being Function of X,(TOP-REAL n) holds <-> f is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f being Function of X,(TOP-REAL n) holds <-> f is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: <-> f is Function of X,(TOP-REAL n) set g = <-> f; A1: dom (<-> f) = dom f by VALUED_2:def_33; A2: dom f = X by FUNCT_2:def_1; for x being set st x in X holds (<-> f) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (<-> f) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (<-> f) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f as Function of X,(TOP-REAL n) ; A4: - (f . x) = - (f . x) ; (<-> f) . x = - (f . x) by A1, A2, VALUED_2:def_33; hence (<-> f) . x in the carrier of (TOP-REAL n) by A4; ::_thesis: verum end; hence <-> f is Function of X,(TOP-REAL n) by A1, A2, FUNCT_2:3; ::_thesis: verum end; theorem Th34: :: TOPREALC:34 for i, n being Nat for f being Function of (TOP-REAL i),(TOP-REAL n) holds f (-) is Function of (TOP-REAL i),(TOP-REAL n) proof let i, n be Nat; ::_thesis: for f being Function of (TOP-REAL i),(TOP-REAL n) holds f (-) is Function of (TOP-REAL i),(TOP-REAL n) set X = the carrier of (TOP-REAL i); let f be Function of the carrier of (TOP-REAL i),(TOP-REAL n); ::_thesis: f (-) is Function of (TOP-REAL i),(TOP-REAL n) set g = f (-) ; A1: dom (f (-)) = dom f by VALUED_2:def_34; A2: dom f = the carrier of (TOP-REAL i) by FUNCT_2:def_1; for x being set st x in the carrier of (TOP-REAL i) holds (f (-)) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in the carrier of (TOP-REAL i) implies (f (-)) . x in the carrier of (TOP-REAL n) ) assume x in the carrier of (TOP-REAL i) ; ::_thesis: (f (-)) . x in the carrier of (TOP-REAL n) then reconsider x = x as Element of the carrier of (TOP-REAL i) ; reconsider y = - x as Element of the carrier of (TOP-REAL i) ; reconsider f = f as Function of the carrier of (TOP-REAL i),(TOP-REAL n) ; (f (-)) . x = f . y by A1, A2, VALUED_2:def_34; hence (f (-)) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f (-) is Function of (TOP-REAL i),(TOP-REAL n) by A1, A2, FUNCT_2:3; ::_thesis: verum end; theorem Th35: :: TOPREALC:35 for X being set for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [+] r is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [+] r is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for r being real number for f being Function of X,(TOP-REAL n) holds f [+] r is Function of X,(TOP-REAL n) let r be real number ; ::_thesis: for f being Function of X,(TOP-REAL n) holds f [+] r is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: f [+] r is Function of X,(TOP-REAL n) set h = f [+] r; dom f = X by FUNCT_2:def_1; then A1: dom (f [+] r) = X by VALUED_2:def_37; for x being set st x in X holds (f [+] r) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f [+] r) . x in the carrier of (TOP-REAL n) ) assume A2: x in X ; ::_thesis: (f [+] r) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A2; reconsider f = f as Function of X,(TOP-REAL n) ; A3: (f . x) + r = (f . x) + r ; (f [+] r) . x = (f . x) + r by A1, VALUED_2:def_37; hence (f [+] r) . x in the carrier of (TOP-REAL n) by A3; ::_thesis: verum end; hence f [+] r is Function of X,(TOP-REAL n) by A1, FUNCT_2:3; ::_thesis: verum end; theorem :: TOPREALC:36 for X being set for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [-] r is Function of X,(TOP-REAL n) by Th35; theorem Th37: :: TOPREALC:37 for X being set for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [#] r is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [#] r is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for r being real number for f being Function of X,(TOP-REAL n) holds f [#] r is Function of X,(TOP-REAL n) let r be real number ; ::_thesis: for f being Function of X,(TOP-REAL n) holds f [#] r is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: f [#] r is Function of X,(TOP-REAL n) set h = f [#] r; dom f = X by FUNCT_2:def_1; then A1: dom (f [#] r) = X by VALUED_2:def_39; for x being set st x in X holds (f [#] r) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f [#] r) . x in the carrier of (TOP-REAL n) ) assume A2: x in X ; ::_thesis: (f [#] r) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A2; reconsider f = f as Function of X,(TOP-REAL n) ; A3: (f . x) (#) r = (f . x) (#) r ; (f [#] r) . x = (f . x) (#) r by A1, VALUED_2:def_39; hence (f [#] r) . x in the carrier of (TOP-REAL n) by A3; ::_thesis: verum end; hence f [#] r is Function of X,(TOP-REAL n) by A1, FUNCT_2:3; ::_thesis: verum end; theorem :: TOPREALC:38 for X being set for n being Nat for r being real number for f being Function of X,(TOP-REAL n) holds f [/] r is Function of X,(TOP-REAL n) by Th37; theorem Th39: :: TOPREALC:39 for X being set for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <++> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <++> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f, g being Function of X,(TOP-REAL n) holds f <++> g is Function of X,(TOP-REAL n) let f, g be Function of X,(TOP-REAL n); ::_thesis: f <++> g is Function of X,(TOP-REAL n) set h = f <++> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <++> g) = X by A1, VALUED_2:def_45; for x being set st x in X holds (f <++> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <++> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <++> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f, g = g as Function of X,(TOP-REAL n) ; A4: (f . x) + (g . x) = (f . x) + (g . x) ; (f <++> g) . x = (f . x) + (g . x) by A2, VALUED_2:def_45; hence (f <++> g) . x in the carrier of (TOP-REAL n) by A4; ::_thesis: verum end; hence f <++> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th40: :: TOPREALC:40 for X being set for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <--> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <--> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f, g being Function of X,(TOP-REAL n) holds f <--> g is Function of X,(TOP-REAL n) let f, g be Function of X,(TOP-REAL n); ::_thesis: f <--> g is Function of X,(TOP-REAL n) set h = f <--> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <--> g) = X by A1, VALUED_2:def_46; for x being set st x in X holds (f <--> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <--> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <--> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f, g = g as Function of X,(TOP-REAL n) ; A4: (f . x) - (g . x) = (f . x) - (g . x) ; (f <--> g) . x = (f . x) - (g . x) by A2, VALUED_2:def_46; hence (f <--> g) . x in the carrier of (TOP-REAL n) by A4; ::_thesis: verum end; hence f <--> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th41: :: TOPREALC:41 for X being set for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <##> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f, g being Function of X,(TOP-REAL n) holds f <##> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f, g being Function of X,(TOP-REAL n) holds f <##> g is Function of X,(TOP-REAL n) let f, g be Function of X,(TOP-REAL n); ::_thesis: f <##> g is Function of X,(TOP-REAL n) set h = f <##> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <##> g) = X by A1, VALUED_2:def_47; for x being set st x in X holds (f <##> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <##> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <##> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f, g = g as Function of X,(TOP-REAL n) ; (f <##> g) . x = (f . x) (#) (g . x) by A2, VALUED_2:def_47; hence (f <##> g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f <##> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th42: :: TOPREALC:42 for X being set for n being Nat for f, g being Function of X,(TOP-REAL n) holds f g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f, g being Function of X,(TOP-REAL n) holds f g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f, g being Function of X,(TOP-REAL n) holds f g is Function of X,(TOP-REAL n) let f, g be Function of X,(TOP-REAL n); ::_thesis: f g is Function of X,(TOP-REAL n) set h = f g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f g) = X by A1, VALUED_2:def_48; for x being set st x in X holds (f g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f, g = g as Function of X,(TOP-REAL n) ; (f g) . x = (f . x) /" (g . x) by A2, VALUED_2:def_48; hence (f g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th43: :: TOPREALC:43 for X being set for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <+> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <+> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <+> g is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: for g being Function of X,R^1 holds f <+> g is Function of X,(TOP-REAL n) let g be Function of X,R^1; ::_thesis: f <+> g is Function of X,(TOP-REAL n) set h = f <+> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <+> g) = X by A1, VALUED_2:def_41; for x being set st x in X holds (f <+> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <+> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <+> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f as Function of X,(TOP-REAL n) ; (f <+> g) . x = (f . x) + (g . x) by A2, VALUED_2:def_41; hence (f <+> g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f <+> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th44: :: TOPREALC:44 for X being set for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <-> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <-> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <-> g is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: for g being Function of X,R^1 holds f <-> g is Function of X,(TOP-REAL n) let g be Function of X,R^1; ::_thesis: f <-> g is Function of X,(TOP-REAL n) set h = f <-> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <-> g) = X by A1, VALUED_2:61; for x being set st x in X holds (f <-> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <-> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <-> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f as Function of X,(TOP-REAL n) ; (f <-> g) . x = (f . x) - (g . x) by A2, VALUED_2:62; hence (f <-> g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f <-> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th45: :: TOPREALC:45 for X being set for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <#> g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <#> g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f <#> g is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: for g being Function of X,R^1 holds f <#> g is Function of X,(TOP-REAL n) let g be Function of X,R^1; ::_thesis: f <#> g is Function of X,(TOP-REAL n) set h = f <#> g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f <#> g) = X by A1, VALUED_2:def_43; for x being set st x in X holds (f <#> g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f <#> g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f <#> g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f as Function of X,(TOP-REAL n) ; (f <#> g) . x = (f . x) (#) (g . x) by A2, VALUED_2:def_43; hence (f <#> g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f <#> g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; theorem Th46: :: TOPREALC:46 for X being set for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f g is Function of X,(TOP-REAL n) proof let X be set ; ::_thesis: for n being Nat for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f g is Function of X,(TOP-REAL n) let n be Nat; ::_thesis: for f being Function of X,(TOP-REAL n) for g being Function of X,R^1 holds f g is Function of X,(TOP-REAL n) let f be Function of X,(TOP-REAL n); ::_thesis: for g being Function of X,R^1 holds f g is Function of X,(TOP-REAL n) let g be Function of X,R^1; ::_thesis: f g is Function of X,(TOP-REAL n) set h = f g; A1: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then A2: dom (f g) = X by A1, VALUED_2:71; for x being set st x in X holds (f g) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in X implies (f g) . x in the carrier of (TOP-REAL n) ) assume A3: x in X ; ::_thesis: (f g) . x in the carrier of (TOP-REAL n) then reconsider X = X as non empty set ; reconsider x = x as Element of X by A3; reconsider f = f as Function of X,(TOP-REAL n) ; (f g) . x = (f . x) (/) (g . x) by A2, VALUED_2:72; hence (f g) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f g is Function of X,(TOP-REAL n) by A2, FUNCT_2:3; ::_thesis: verum end; definition let n be Nat; let T be non empty set ; let R be real-membered set ; let f be Function of T,R; func incl (f,n) -> Function of T,(TOP-REAL n) means :Def4: :: TOPREALC:def 4 for t being Element of T holds it . t = n |-> (f . t); existence ex b1 being Function of T,(TOP-REAL n) st for t being Element of T holds b1 . t = n |-> (f . t) proof defpred S1[ set , set ] means \$2 = n |-> (f . \$1); A1: for x being Element of T ex y being Element of (TOP-REAL n) st S1[x,y] proof let x be Element of T; ::_thesis: ex y being Element of (TOP-REAL n) st S1[x,y] f . x in REAL by XREAL_0:def_1; then n |-> (f . x) is Element of REAL n by FINSEQ_2:112; then reconsider y = n |-> (f . x) as Point of (TOP-REAL n) by EUCLID:22; take y ; ::_thesis: S1[x,y] thus S1[x,y] ; ::_thesis: verum end; ex F being Function of T,(TOP-REAL n) st for x being Element of T holds S1[x,F . x] from FUNCT_2:sch_3(A1); hence ex b1 being Function of T,(TOP-REAL n) st for t being Element of T holds b1 . t = n |-> (f . t) ; ::_thesis: verum end; uniqueness for b1, b2 being Function of T,(TOP-REAL n) st ( for t being Element of T holds b1 . t = n |-> (f . t) ) & ( for t being Element of T holds b2 . t = n |-> (f . t) ) holds b1 = b2 proof let F, G be Function of T,(TOP-REAL n); ::_thesis: ( ( for t being Element of T holds F . t = n |-> (f . t) ) & ( for t being Element of T holds G . t = n |-> (f . t) ) implies F = G ) assume that A2: for t being Element of T holds F . t = n |-> (f . t) and A3: for t being Element of T holds G . t = n |-> (f . t) ; ::_thesis: F = G let t be Element of T; :: according to FUNCT_2:def_8 ::_thesis: F . t = G . t thus F . t = n |-> (f . t) by A2 .= G . t by A3 ; ::_thesis: verum end; end; :: deftheorem Def4 defines incl TOPREALC:def_4_:_ for n being Nat for T being non empty set for R being real-membered set for f being Function of T,R for b5 being Function of T,(TOP-REAL n) holds ( b5 = incl (f,n) iff for t being Element of T holds b5 . t = n |-> (f . t) ); theorem Th47: :: TOPREALC:47 for x being set for n being Nat for T being non empty TopSpace for R being real-membered set for f being Function of T,R for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t proof let x be set ; ::_thesis: for n being Nat for T being non empty TopSpace for R being real-membered set for f being Function of T,R for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t let n be Nat; ::_thesis: for T being non empty TopSpace for R being real-membered set for f being Function of T,R for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t let T be non empty TopSpace; ::_thesis: for R being real-membered set for f being Function of T,R for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t let R be real-membered set ; ::_thesis: for f being Function of T,R for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t let f be Function of T,R; ::_thesis: for t being Point of T st x in Seg n holds ((incl (f,n)) . t) . x = f . t let t be Point of T; ::_thesis: ( x in Seg n implies ((incl (f,n)) . t) . x = f . t ) assume A1: x in Seg n ; ::_thesis: ((incl (f,n)) . t) . x = f . t thus ((incl (f,n)) . t) . x = (n |-> (f . t)) . x by Def4 .= f . t by A1, FINSEQ_2:57 ; ::_thesis: verum end; theorem Th48: :: TOPREALC:48 for T being non empty set for R being real-membered set for f being Function of T,R holds incl (f,0) = T --> 0 proof let T be non empty set ; ::_thesis: for R being real-membered set for f being Function of T,R holds incl (f,0) = T --> 0 let R be real-membered set ; ::_thesis: for f being Function of T,R holds incl (f,0) = T --> 0 let f be Function of T,R; ::_thesis: incl (f,0) = T --> 0 reconsider z = 0 as Element of (TOP-REAL 0) ; incl (f,0) = T --> z proof let x be Element of T; :: according to FUNCT_2:def_8 ::_thesis: (incl (f,0)) . x = (T --> z) . x thus (incl (f,0)) . x = (T --> z) . x ; ::_thesis: verum end; hence incl (f,0) = T --> 0 ; ::_thesis: verum end; theorem Th49: :: TOPREALC:49 for n being Nat for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <+> g = f <++> (incl (g,n)) proof let n be Nat; ::_thesis: for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <+> g = f <++> (incl (g,n)) let T be non empty TopSpace; ::_thesis: for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <+> g = f <++> (incl (g,n)) let f be Function of T,(TOP-REAL n); ::_thesis: for g being Function of T,R^1 holds f <+> g = f <++> (incl (g,n)) let g be Function of T,R^1; ::_thesis: f <+> g = f <++> (incl (g,n)) set I = incl (g,n); reconsider h = f <+> g as Function of T,(TOP-REAL n) by Th43; reconsider G = f <++> (incl (g,n)) as Function of T,(TOP-REAL n) by Th39; h = G proof let t be Point of T; :: according to FUNCT_2:def_8 ::_thesis: h . t = G . t A1: dom h = the carrier of T by FUNCT_2:def_1; A2: (f . t) + ((incl (g,n)) . t) = (f . t) + (g . t) proof A3: ( dom (f . t) = Seg n & dom ((incl (g,n)) . t) = Seg n ) by FINSEQ_1:89; A4: dom ((f . t) + ((incl (g,n)) . t)) = (dom (f . t)) /\ (dom ((incl (g,n)) . t)) by VALUED_1:def_1 .= Seg n by A3 ; A5: dom ((f . t) + (g . t)) = dom (f . t) by VALUED_1:def_2; thus dom ((f . t) + ((incl (g,n)) . t)) = dom ((f . t) + (g . t)) by A4, FINSEQ_1:89; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f . t) + ((incl (g,n)) . t)) or ((f . t) + ((incl (g,n)) . t)) . b1 = ((f . t) + (g . t)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f . t) + ((incl (g,n)) . t)) or ((f . t) + ((incl (g,n)) . t)) . x = ((f . t) + (g . t)) . x ) assume A6: x in dom ((f . t) + ((incl (g,n)) . t)) ; ::_thesis: ((f . t) + ((incl (g,n)) . t)) . x = ((f . t) + (g . t)) . x hence ((f . t) + ((incl (g,n)) . t)) . x = ((f . t) . x) + (((incl (g,n)) . t) . x) by VALUED_1:def_1 .= ((f . t) . x) + (g . t) by A4, A6, Th47 .= ((f . t) + (g . t)) . x by A3, A4, A6, A5, VALUED_1:def_2 ; ::_thesis: verum end; dom G = the carrier of T by FUNCT_2:def_1; hence G . t = (f . t) + ((incl (g,n)) . t) by VALUED_2:def_45 .= h . t by A1, A2, VALUED_2:def_41 ; ::_thesis: verum end; hence f <+> g = f <++> (incl (g,n)) ; ::_thesis: verum end; theorem Th50: :: TOPREALC:50 for n being Nat for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <-> g = f <--> (incl (g,n)) proof let n be Nat; ::_thesis: for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <-> g = f <--> (incl (g,n)) let T be non empty TopSpace; ::_thesis: for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <-> g = f <--> (incl (g,n)) let f be Function of T,(TOP-REAL n); ::_thesis: for g being Function of T,R^1 holds f <-> g = f <--> (incl (g,n)) let g be Function of T,R^1; ::_thesis: f <-> g = f <--> (incl (g,n)) set I = incl (g,n); reconsider h = f <-> g as Function of T,(TOP-REAL n) by Th44; reconsider G = f <--> (incl (g,n)) as Function of T,(TOP-REAL n) by Th40; h = G proof let t be Point of T; :: according to FUNCT_2:def_8 ::_thesis: h . t = G . t A1: dom h = the carrier of T by FUNCT_2:def_1; A2: (f . t) - ((incl (g,n)) . t) = (f . t) - (g . t) proof A3: ( dom (f . t) = Seg n & dom ((incl (g,n)) . t) = Seg n ) by FINSEQ_1:89; A4: dom ((f . t) - ((incl (g,n)) . t)) = (dom (f . t)) /\ (dom ((incl (g,n)) . t)) by VALUED_1:12 .= Seg n by A3 ; A5: dom ((f . t) - (g . t)) = dom (f . t) by VALUED_1:def_2; thus dom ((f . t) - ((incl (g,n)) . t)) = dom ((f . t) - (g . t)) by A4, FINSEQ_1:89; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f . t) - ((incl (g,n)) . t)) or ((f . t) - ((incl (g,n)) . t)) . b1 = ((f . t) - (g . t)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f . t) - ((incl (g,n)) . t)) or ((f . t) - ((incl (g,n)) . t)) . x = ((f . t) - (g . t)) . x ) assume A6: x in dom ((f . t) - ((incl (g,n)) . t)) ; ::_thesis: ((f . t) - ((incl (g,n)) . t)) . x = ((f . t) - (g . t)) . x hence ((f . t) - ((incl (g,n)) . t)) . x = ((f . t) . x) - (((incl (g,n)) . t) . x) by VALUED_1:13 .= ((f . t) . x) - (g . t) by A4, A6, Th47 .= ((f . t) - (g . t)) . x by A3, A4, A6, A5, VALUED_1:def_2 ; ::_thesis: verum end; dom G = the carrier of T by FUNCT_2:def_1; hence G . t = (f . t) - ((incl (g,n)) . t) by VALUED_2:def_46 .= h . t by A1, A2, VALUED_2:62 ; ::_thesis: verum end; hence f <-> g = f <--> (incl (g,n)) ; ::_thesis: verum end; theorem Th51: :: TOPREALC:51 for n being Nat for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <#> g = f <##> (incl (g,n)) proof let n be Nat; ::_thesis: for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <#> g = f <##> (incl (g,n)) let T be non empty TopSpace; ::_thesis: for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f <#> g = f <##> (incl (g,n)) let f be Function of T,(TOP-REAL n); ::_thesis: for g being Function of T,R^1 holds f <#> g = f <##> (incl (g,n)) let g be Function of T,R^1; ::_thesis: f <#> g = f <##> (incl (g,n)) set I = incl (g,n); reconsider h = f <#> g as Function of T,(TOP-REAL n) by Th45; reconsider G = f <##> (incl (g,n)) as Function of T,(TOP-REAL n) by Th41; h = G proof let t be Point of T; :: according to FUNCT_2:def_8 ::_thesis: h . t = G . t A1: dom h = the carrier of T by FUNCT_2:def_1; A2: (f . t) (#) ((incl (g,n)) . t) = (f . t) (#) (g . t) proof A3: ( dom (f . t) = Seg n & dom ((incl (g,n)) . t) = Seg n ) by FINSEQ_1:89; A4: dom ((f . t) (#) ((incl (g,n)) . t)) = (dom (f . t)) /\ (dom ((incl (g,n)) . t)) by VALUED_1:def_4 .= Seg n by A3 ; hence dom ((f . t) (#) ((incl (g,n)) . t)) = dom ((f . t) (#) (g . t)) by FINSEQ_1:89; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f . t) (#) ((incl (g,n)) . t)) or ((f . t) (#) ((incl (g,n)) . t)) . b1 = ((f . t) (#) (g . t)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f . t) (#) ((incl (g,n)) . t)) or ((f . t) (#) ((incl (g,n)) . t)) . x = ((f . t) (#) (g . t)) . x ) assume A5: x in dom ((f . t) (#) ((incl (g,n)) . t)) ; ::_thesis: ((f . t) (#) ((incl (g,n)) . t)) . x = ((f . t) (#) (g . t)) . x hence ((f . t) (#) ((incl (g,n)) . t)) . x = ((f . t) . x) * (((incl (g,n)) . t) . x) by VALUED_1:def_4 .= ((f . t) . x) * (g . t) by A4, A5, Th47 .= ((f . t) (#) (g . t)) . x by VALUED_1:6 ; ::_thesis: verum end; dom G = the carrier of T by FUNCT_2:def_1; hence G . t = (f . t) (#) ((incl (g,n)) . t) by VALUED_2:def_47 .= h . t by A1, A2, VALUED_2:def_43 ; ::_thesis: verum end; hence f <#> g = f <##> (incl (g,n)) ; ::_thesis: verum end; theorem :: TOPREALC:52 for n being Nat for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f g = f (incl (g,n)) proof let n be Nat; ::_thesis: for T being non empty TopSpace for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f g = f (incl (g,n)) let T be non empty TopSpace; ::_thesis: for f being Function of T,(TOP-REAL n) for g being Function of T,R^1 holds f g = f (incl (g,n)) let f be Function of T,(TOP-REAL n); ::_thesis: for g being Function of T,R^1 holds f g = f (incl (g,n)) let g be Function of T,R^1; ::_thesis: f g = f (incl (g,n)) set I = incl (g,n); reconsider h = f g as Function of T,(TOP-REAL n) by Th46; reconsider G = f (incl (g,n)) as Function of T,(TOP-REAL n) by Th42; h = G proof let t be Point of T; :: according to FUNCT_2:def_8 ::_thesis: h . t = G . t A1: dom h = the carrier of T by FUNCT_2:def_1; A2: (f . t) /" ((incl (g,n)) . t) = (f . t) (#) ((g ") . t) proof A3: ( dom (f . t) = Seg n & dom ((incl (g,n)) . t) = Seg n ) by FINSEQ_1:89; A4: dom ((f . t) /" ((incl (g,n)) . t)) = (dom (f . t)) /\ (dom ((incl (g,n)) . t)) by VALUED_1:16 .= Seg n by A3 ; hence dom ((f . t) /" ((incl (g,n)) . t)) = dom ((f . t) (#) ((g ") . t)) by FINSEQ_1:89; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds ( not b1 in dom ((f . t) /" ((incl (g,n)) . t)) or ((f . t) /" ((incl (g,n)) . t)) . b1 = ((f . t) (#) ((g ") . t)) . b1 ) let x be set ; ::_thesis: ( not x in dom ((f . t) /" ((incl (g,n)) . t)) or ((f . t) /" ((incl (g,n)) . t)) . x = ((f . t) (#) ((g ") . t)) . x ) assume A5: x in dom ((f . t) /" ((incl (g,n)) . t)) ; ::_thesis: ((f . t) /" ((incl (g,n)) . t)) . x = ((f . t) (#) ((g ") . t)) . x thus ((f . t) /" ((incl (g,n)) . t)) . x = ((f . t) . x) / (((incl (g,n)) . t) . x) by VALUED_1:17 .= ((f . t) . x) / (g . t) by A4, A5, Th47 .= ((f . t) . x) * ((g ") . t) by VALUED_1:10 .= ((f . t) (#) ((g ") . t)) . x by VALUED_1:6 ; ::_thesis: verum end; dom G = the carrier of T by FUNCT_2:def_1; hence G . t = (f . t) /" ((incl (g,n)) . t) by VALUED_2:def_48 .= h . t by A1, A2, VALUED_2:def_43 ; ::_thesis: verum end; hence f g = f (incl (g,n)) ; ::_thesis: verum end; definition let n be Nat; set T = TOP-REAL n; set c = the carrier of (TOP-REAL n); A1: the carrier of [:(TOP-REAL n),(TOP-REAL n):] = [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] by BORSUK_1:def_2; func TIMES n -> Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n) means :Def5: :: TOPREALC:def 5 for x, y being Point of (TOP-REAL n) holds it . (x,y) = x (#) y; existence ex b1 being Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n) st for x, y being Point of (TOP-REAL n) holds b1 . (x,y) = x (#) y proof deffunc H1( Point of (TOP-REAL n), Point of (TOP-REAL n)) -> Point of (TOP-REAL n) = \$1 (#) \$2; ex f being Function of [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n) st for x, y being Element of the carrier of (TOP-REAL n) holds f . (x,y) = H1(x,y) from BINOP_1:sch_4(); hence ex b1 being Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n) st for x, y being Point of (TOP-REAL n) holds b1 . (x,y) = x (#) y by A1; ::_thesis: verum end; uniqueness for b1, b2 being Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n) st ( for x, y being Point of (TOP-REAL n) holds b1 . (x,y) = x (#) y ) & ( for x, y being Point of (TOP-REAL n) holds b2 . (x,y) = x (#) y ) holds b1 = b2 proof let f, g be Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n); ::_thesis: ( ( for x, y being Point of (TOP-REAL n) holds f . (x,y) = x (#) y ) & ( for x, y being Point of (TOP-REAL n) holds g . (x,y) = x (#) y ) implies f = g ) assume that A2: for x, y being Point of (TOP-REAL n) holds f . (x,y) = x (#) y and A3: for x, y being Point of (TOP-REAL n) holds g . (x,y) = x (#) y ; ::_thesis: f = g now__::_thesis:_for_x,_y_being_Point_of_(TOP-REAL_n)_holds_f_._(x,y)_=_g_._(x,y) let x, y be Point of (TOP-REAL n); ::_thesis: f . (x,y) = g . (x,y) thus f . (x,y) = x (#) y by A2 .= g . (x,y) by A3 ; ::_thesis: verum end; hence f = g by A1, BINOP_1:2; ::_thesis: verum end; end; :: deftheorem Def5 defines TIMES TOPREALC:def_5_:_ for n being Nat for b2 being Function of [:(TOP-REAL n),(TOP-REAL n):],(TOP-REAL n) holds ( b2 = TIMES n iff for x, y being Point of (TOP-REAL n) holds b2 . (x,y) = x (#) y ); theorem Th53: :: TOPREALC:53 TIMES 0 = [:(TOP-REAL 0),(TOP-REAL 0):] --> (0. (TOP-REAL 0)) proof set T = TOP-REAL 0; let x be Element of the carrier of [:(TOP-REAL 0),(TOP-REAL 0):]; :: according to FUNCT_2:def_8 ::_thesis: (TIMES 0) . x = ([:(TOP-REAL 0),(TOP-REAL 0):] --> (0. (TOP-REAL 0))) . x thus (TIMES 0) . x = ([:(TOP-REAL 0),(TOP-REAL 0):] --> (0. (TOP-REAL 0))) . x ; ::_thesis: verum end; theorem Th54: :: TOPREALC:54 for n being Nat for T being non empty TopSpace for f, g being Function of T,(TOP-REAL n) holds f <##> g = (TIMES n) .: (f,g) proof let n be Nat; ::_thesis: for T being non empty TopSpace for f, g being Function of T,(TOP-REAL n) holds f <##> g = (TIMES n) .: (f,g) let T be non empty TopSpace; ::_thesis: for f, g being Function of T,(TOP-REAL n) holds f <##> g = (TIMES n) .: (f,g) set R = TOP-REAL n; set F = TIMES n; let f, g be Function of T,(TOP-REAL n); ::_thesis: f <##> g = (TIMES n) .: (f,g) A1: dom (f <##> g) = (dom f) /\ (dom g) by VALUED_2:def_47; dom (TIMES n) = the carrier of [:(TOP-REAL n),(TOP-REAL n):] by FUNCT_2:def_1 .= [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] by BORSUK_1:def_2 ; then [:(rng f),(rng g):] c= dom (TIMES n) by ZFMISC_1:96; then A2: dom ((TIMES n) .: (f,g)) = (dom f) /\ (dom g) by FUNCOP_1:69; now__::_thesis:_for_x_being_set_st_x_in_dom_(f_<##>_g)_holds_ (f_<##>_g)_._x_=_((TIMES_n)_.:_(f,g))_._x let x be set ; ::_thesis: ( x in dom (f <##> g) implies (f <##> g) . x = ((TIMES n) .: (f,g)) . x ) assume A3: x in dom (f <##> g) ; ::_thesis: (f <##> g) . x = ((TIMES n) .: (f,g)) . x then A4: ( f . x is Point of (TOP-REAL n) & g . x is Point of (TOP-REAL n) ) by FUNCT_2:5; thus (f <##> g) . x = (f . x) (#) (g . x) by A3, VALUED_2:def_47 .= (TIMES n) . ((f . x),(g . x)) by A4, Def5 .= ((TIMES n) .: (f,g)) . x by A1, A2, A3, FUNCOP_1:22 ; ::_thesis: verum end; hence f <##> g = (TIMES n) .: (f,g) by A1, A2, FUNCT_1:2; ::_thesis: verum end; definition let m, n be Nat; A1: ( m in NAT & n in NAT ) by ORDINAL1:def_12; func PROJ (m,n) -> Function of (TOP-REAL m),R^1 means :Def6: :: TOPREALC:def 6 for p being Element of (TOP-REAL m) holds it . p = p /. n; existence ex b1 being Function of (TOP-REAL m),R^1 st for p being Element of (TOP-REAL m) holds b1 . p = p /. n by A1, JORDAN2B:1; uniqueness for b1, b2 being Function of (TOP-REAL m),R^1 st ( for p being Element of (TOP-REAL m) holds b1 . p = p /. n ) & ( for p being Element of (TOP-REAL m) holds b2 . p = p /. n ) holds b1 = b2 proof let F, G be Function of (TOP-REAL m),R^1; ::_thesis: ( ( for p being Element of (TOP-REAL m) holds F . p = p /. n ) & ( for p being Element of (TOP-REAL m) holds G . p = p /. n ) implies F = G ) assume that A2: for p being Element of (TOP-REAL m) holds F . p = p /. n and A3: for p being Element of (TOP-REAL m) holds G . p = p /. n ; ::_thesis: F = G let p be Element of (TOP-REAL m); :: according to FUNCT_2:def_8 ::_thesis: F . p = G . p thus F . p = p /. n by A2 .= G . p by A3 ; ::_thesis: verum end; end; :: deftheorem Def6 defines PROJ TOPREALC:def_6_:_ for m, n being Nat for b3 being Function of (TOP-REAL m),R^1 holds ( b3 = PROJ (m,n) iff for p being Element of (TOP-REAL m) holds b3 . p = p /. n ); theorem Th55: :: TOPREALC:55 for m, n being Nat for r being real number for p being Point of (TOP-REAL m) st n in dom p holds (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ proof let m, n be Nat; ::_thesis: for r being real number for p being Point of (TOP-REAL m) st n in dom p holds (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ let r be real number ; ::_thesis: for p being Point of (TOP-REAL m) st n in dom p holds (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ let p be Point of (TOP-REAL m); ::_thesis: ( n in dom p implies (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ ) assume A1: n in dom p ; ::_thesis: (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ A2: m in NAT by ORDINAL1:def_12; percases ( r <= 0 or 0 < r ) ; supposeA3: r <= 0 ; ::_thesis: (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ then ].((p /. n) - r),((p /. n) + r).[ = {} ; hence (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ by A3; ::_thesis: verum end; supposeA4: 0 < r ; ::_thesis: (PROJ (m,n)) .: (Ball (p,r)) = ].((p /. n) - r),((p /. n) + r).[ A5: dom p = Seg m by FINSEQ_1:89; thus (PROJ (m,n)) .: (Ball (p,r)) c= ].((p /. n) - r),((p /. n) + r).[ :: according to XBOOLE_0:def_10 ::_thesis: ].((p /. n) - r),((p /. n) + r).[ c= (PROJ (m,n)) .: (Ball (p,r)) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (PROJ (m,n)) .: (Ball (p,r)) or y in ].((p /. n) - r),((p /. n) + r).[ ) assume y in (PROJ (m,n)) .: (Ball (p,r)) ; ::_thesis: y in ].((p /. n) - r),((p /. n) + r).[ then consider x being Element of (TOP-REAL m) such that A6: x in Ball (p,r) and A7: y = (PROJ (m,n)) . x by FUNCT_2:65; A8: (PROJ (m,n)) . x = x /. n by Def6; A9: |.(x - p).| < r by A2, A6, TOPREAL9:7; 0 <= Sum (sqr (x - p)) by RVSUM_1:86; then (sqrt (Sum (sqr (x - p)))) ^2 = Sum (sqr (x - p)) by SQUARE_1:def_2; then A10: Sum (sqr (x - p)) < r ^2 by A9, SQUARE_1:16; dom x = Seg m by FINSEQ_1:89; then ((x /. n) - (p /. n)) ^2 <= Sum (sqr (x - p)) by A5, A1, EUCLID_9:8; then ((x /. n) - (p /. n)) ^2 < r ^2 by A10, XXREAL_0:2; then ( - r < (x /. n) - (p /. n) & (x /. n) - (p /. n) < r ) by A4, SQUARE_1:48; then ( (- r) + (p /. n) < ((x /. n) - (p /. n)) + (p /. n) & ((x /. n) - (p /. n)) + (p /. n) < r + (p /. n) ) by XREAL_1:6; hence y in ].((p /. n) - r),((p /. n) + r).[ by A7, A8, XXREAL_1:4; ::_thesis: verum end; let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in ].((p /. n) - r),((p /. n) + r).[ or y in (PROJ (m,n)) .: (Ball (p,r)) ) assume A11: y in ].((p /. n) - r),((p /. n) + r).[ ; ::_thesis: y in (PROJ (m,n)) .: (Ball (p,r)) then reconsider y = y as Element of REAL ; set x = p +* (n,y); reconsider X = p +* (n,y) as FinSequence of REAL by EUCLID:24; A12: dom X = dom p by FUNCT_7:30; A13: p /. n = p . n by A1, PARTFUN1:def_6; ( (p /. n) - r < y & y < (p /. n) + r ) by A11, XXREAL_1:4; then A14: ( y - (p /. n) < r & - r < y - (p /. n) ) by XREAL_1:19, XREAL_1:20; (p +* (n,y)) - p = (0* m) +* (n,(y - (p . n))) by Th17; then |.((p +* (n,y)) - p).| = abs (y - (p . n)) by A1, A5, Th13; then |.((p +* (n,y)) - p).| < r by A13, A14, SEQ_2:1; then A15: p +* (n,y) in Ball (p,r) by A2, TOPREAL9:7; (PROJ (m,n)) . (p +* (n,y)) = (p +* (n,y)) /. n by Def6 .= X . n by A12, A1, PARTFUN1:def_6 .= y by A1, FUNCT_7:31 ; hence y in (PROJ (m,n)) .: (Ball (p,r)) by A15, FUNCT_2:35; ::_thesis: verum end; end; end; theorem :: TOPREALC:56 for T being non empty TopSpace for m being non zero Nat for f being Function of T,R^1 holds f = (PROJ (m,m)) * (incl (f,m)) proof let T be non empty TopSpace; ::_thesis: for m being non zero Nat for f being Function of T,R^1 holds f = (PROJ (m,m)) * (incl (f,m)) let m be non zero Nat; ::_thesis: for f being Function of T,R^1 holds f = (PROJ (m,m)) * (incl (f,m)) let f be Function of T,R^1; ::_thesis: f = (PROJ (m,m)) * (incl (f,m)) let p be Point of T; :: according to FUNCT_2:def_8 ::_thesis: f . p = ((PROJ (m,m)) * (incl (f,m))) . p set I = incl (f,m); reconsider G = m |-> (f . p) as FinSequence of REAL ; A1: m in NAT by ORDINAL1:def_12; 1 <= m by NAT_1:14; then A2: m in Seg m by A1; A3: dom (m |-> (f . p)) = Seg m by FUNCOP_1:13; thus ((PROJ (m,m)) * (incl (f,m))) . p = (PROJ (m,m)) . ((incl (f,m)) . p) by FUNCT_2:15 .= ((incl (f,m)) . p) /. m by Def6 .= G /. m by Def4 .= G . m by A2, A3, PARTFUN1:def_6 .= f . p by A2, FINSEQ_2:57 ; ::_thesis: verum end; begin registration let T be non empty TopSpace; cluster Relation-like non-empty the carrier of T -defined the carrier of R^1 -valued Function-like non empty total V18( the carrier of T, the carrier of R^1) complex-valued ext-real-valued real-valued continuous for Element of bool [: the carrier of T, the carrier of R^1:]; existence ex b1 being Function of T,R^1 st ( b1 is non-empty & b1 is continuous ) proof take T --> (R^1 1) ; ::_thesis: ( T --> (R^1 1) is non-empty & T --> (R^1 1) is continuous ) thus ( T --> (R^1 1) is non-empty & T --> (R^1 1) is continuous ) ; ::_thesis: verum end; end; theorem :: TOPREALC:57 for n, m being Nat st n in Seg m holds PROJ (m,n) is continuous proof let n, m be Nat; ::_thesis: ( n in Seg m implies PROJ (m,n) is continuous ) assume A1: n in Seg m ; ::_thesis: PROJ (m,n) is continuous A2: m in NAT by ORDINAL1:def_12; for p being Element of (TOP-REAL m) holds (PROJ (m,n)) . p = p /. n by Def6; hence PROJ (m,n) is continuous by A2, A1, JORDAN2B:18; ::_thesis: verum end; theorem :: TOPREALC:58 for n, m being Nat st n in Seg m holds PROJ (m,n) is open proof let n, m be Nat; ::_thesis: ( n in Seg m implies PROJ (m,n) is open ) set f = PROJ (m,n); assume A1: n in Seg m ; ::_thesis: PROJ (m,n) is open for p being Point of (TOP-REAL m) for r being positive real number ex s being positive real number st ].(((PROJ (m,n)) . p) - s),(((PROJ (m,n)) . p) + s).[ c= (PROJ (m,n)) .: (Ball (p,r)) proof let p be Point of (TOP-REAL m); ::_thesis: for r being positive real number ex s being positive real number st ].(((PROJ (m,n)) . p) - s),(((PROJ (m,n)) . p) + s).[ c= (PROJ (m,n)) .: (Ball (p,r)) let r be positive real number ; ::_thesis: ex s being positive real number st ].(((PROJ (m,n)) . p) - s),(((PROJ (m,n)) . p) + s).[ c= (PROJ (m,n)) .: (Ball (p,r)) take r ; ::_thesis: ].(((PROJ (m,n)) . p) - r),(((PROJ (m,n)) . p) + r).[ c= (PROJ (m,n)) .: (Ball (p,r)) A2: dom p = Seg m by FINSEQ_1:89; p /. n = (PROJ (m,n)) . p by Def6; hence ].(((PROJ (m,n)) . p) - r),(((PROJ (m,n)) . p) + r).[ c= (PROJ (m,n)) .: (Ball (p,r)) by A2, A1, Th55; ::_thesis: verum end; hence PROJ (m,n) is open by TOPS_4:13; ::_thesis: verum end; registration let n be Nat; let T be non empty TopSpace; let f be continuous Function of T,R^1; cluster incl (f,n) -> continuous ; coherence incl (f,n) is continuous proof reconsider n = n as Element of NAT by ORDINAL1:def_12; set g = incl (f,n); percases ( n = 0 or n > 0 ) ; supposeA1: n = 0 ; ::_thesis: incl (f,n) is continuous then reconsider z = 0 as Element of (TOP-REAL n) ; incl (f,0) = T --> z by Th48; hence incl (f,n) is continuous by A1; ::_thesis: verum end; supposeA2: n > 0 ; ::_thesis: incl (f,n) is continuous for p being Point of T for V being Subset of (TOP-REAL n) st (incl (f,n)) . p in V & V is open holds ex W being Subset of T st ( p in W & W is open & (incl (f,n)) .: W c= V ) proof let p be Point of T; ::_thesis: for V being Subset of (TOP-REAL n) st (incl (f,n)) . p in V & V is open holds ex W being Subset of T st ( p in W & W is open & (incl (f,n)) .: W c= V ) let V be Subset of (TOP-REAL n); ::_thesis: ( (incl (f,n)) . p in V & V is open implies ex W being Subset of T st ( p in W & W is open & (incl (f,n)) .: W c= V ) ) assume that A3: (incl (f,n)) . p in V and A4: V is open ; ::_thesis: ex W being Subset of T st ( p in W & W is open & (incl (f,n)) .: W c= V ) A5: (incl (f,n)) . p in Int V by A3, A4, TOPS_1:23; reconsider s = (incl (f,n)) . p as Point of (Euclid n) by EUCLID:67; consider r being real number such that A6: r > 0 and A7: Ball (s,r) c= V by A5, GOBOARD6:5; reconsider G = n |-> (f . p) as FinSequence of REAL ; 1 <= n by A2, NAT_1:14; then A8: n in Seg n ; reconsider F = (incl (f,n)) . p as FinSequence of REAL by EUCLID:24; A9: F = n |-> (f . p) by Def4; A10: dom (n |-> (f . p)) = Seg n by FUNCOP_1:13; A11: ((incl (f,n)) . p) /. n = G /. n by Def4 .= G . n by A8, A10, PARTFUN1:def_6 .= f . p by A8, FINSEQ_2:57 ; set R = r / (sqrt n); set RR = R^1 ].((((incl (f,n)) . p) /. n) - (r / (sqrt n))),((((incl (f,n)) . p) /. n) + (r / (sqrt n))).[; f . p in R^1 ].((((incl (f,n)) . p) /. n) - (r / (sqrt n))),((((incl (f,n)) . p) /. n) + (r / (sqrt n))).[ by A2, A11, A6, TOPREAL6:15; then consider W being Subset of T such that A12: ( p in W & W is open ) and A13: f .: W c= R^1 ].((((incl (f,n)) . p) /. n) - (r / (sqrt n))),((((incl (f,n)) . p) /. n) + (r / (sqrt n))).[ by JGRAPH_2:10; take W ; ::_thesis: ( p in W & W is open & (incl (f,n)) .: W c= V ) thus ( p in W & W is open ) by A12; ::_thesis: (incl (f,n)) .: W c= V let y be Element of (TOP-REAL n); :: according to LATTICE7:def_1 ::_thesis: ( not y in (incl (f,n)) .: W or y in V ) assume y in (incl (f,n)) .: W ; ::_thesis: y in V then consider x being Element of T such that A14: x in W and A15: (incl (f,n)) . x = y by FUNCT_2:65; reconsider y1 = y as Point of (Euclid n) by EUCLID:67; reconsider y2 = y1, s2 = s as Element of REAL n ; A16: y2 = n |-> (f . x) by A15, Def4; A17: (Pitag_dist n) . (y1,s) = |.(y2 - s2).| by EUCLID:def_6 .= (sqrt n) * (abs ((f . x) - (f . p))) by A16, A9, Th11 ; f . x in f .: W by A14, FUNCT_2:35; then abs ((f . x) - (f . p)) < r / (sqrt n) by A13, A11, RCOMP_1:1; then dist (y1,s) < r by A2, A17, XREAL_1:79; then y in Ball (s,r) by METRIC_1:11; hence y in V by A7; ::_thesis: verum end; hence incl (f,n) is continuous by JGRAPH_2:10; ::_thesis: verum end; end; end; end; registration let n be Nat; cluster TIMES n -> continuous ; coherence TIMES n is continuous proof percases ( not n is zero or n is zero ) ; suppose not n is zero ; ::_thesis: TIMES n is continuous then reconsider n = n as non zero Element of NAT by ORDINAL1:def_12; set T = TOP-REAL n; set F = TIMES n; set J = (Seg n) --> R^1; set c = the carrier of (TOP-REAL n); A1: TopSpaceMetr (Euclid n) = product ((Seg n) --> R^1) by EUCLID_9:28; A2: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8; then reconsider F = TIMES n as Function of [:(TOP-REAL n),(TOP-REAL n):],(product ((Seg n) --> R^1)) by EUCLID_9:28; A3: the carrier of (TOP-REAL n) = REAL n by EUCLID:22; now__::_thesis:_for_i_being_Element_of_Seg_n_holds_(proj_(((Seg_n)_-->_R^1),i))_*_F_is_continuous let i be Element of Seg n; ::_thesis: (proj (((Seg n) --> R^1),i)) * F is continuous set P = proj (((Seg n) --> R^1),i); A4: ((Seg n) --> R^1) . i = R^1 by FUNCOP_1:7; reconsider f = (proj (((Seg n) --> R^1),i)) * F as Function of [:(TOP-REAL n),(TOP-REAL n):],R^1 by FUNCOP_1:7; A5: the carrier of [:(TOP-REAL n),(TOP-REAL n):] = [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] by BORSUK_1:def_2; A6: for a, b being Point of (TOP-REAL n) holds f . (a,b) = (a . i) * (b . i) proof let a, b be Point of (TOP-REAL n); ::_thesis: f . (a,b) = (a . i) * (b . i) thus f . (a,b) = (proj (((Seg n) --> R^1),i)) . (F . (a,b)) by A5, BINOP_1:18 .= (proj (((Seg n) --> R^1),i)) . (a (#) b) by Def5 .= (a (#) b) . i by A1, A2, YELLOW17:8 .= (a . i) * (b . i) by VALUED_1:5 ; ::_thesis: verum end; deffunc H1( Element of the carrier of (TOP-REAL n), Element of the carrier of (TOP-REAL n)) -> Element of REAL = n . i; consider f1 being Function of [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):],REAL such that A7: for a, b being Element of the carrier of (TOP-REAL n) holds f1 . (a,b) = H1(a,b) from BINOP_1:sch_4(); reconsider f1 = f1 as Function of [:(TOP-REAL n),(TOP-REAL n):],R^1 by A5; deffunc H2( Element of the carrier of (TOP-REAL n), Element of the carrier of (TOP-REAL n)) -> Element of REAL = c2 . i; consider f2 being Function of [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):],REAL such that A8: for a, b being Element of the carrier of (TOP-REAL n) holds f2 . (a,b) = H2(a,b) from BINOP_1:sch_4(); reconsider f1 = f1, f2 = f2 as Function of [:(TOP-REAL n),(TOP-REAL n):],R^1 by A5; for p being Point of [:(TOP-REAL n),(TOP-REAL n):] for r being positive real number ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f1 .: W c= ].((f1 . p) - r),((f1 . p) + r).[ ) proof let p be Point of [:(TOP-REAL n),(TOP-REAL n):]; ::_thesis: for r being positive real number ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f1 .: W c= ].((f1 . p) - r),((f1 . p) + r).[ ) let r be positive real number ; ::_thesis: ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f1 .: W c= ].((f1 . p) - r),((f1 . p) + r).[ ) consider p1, p2 being Point of (TOP-REAL n) such that A9: p = [p1,p2] by BORSUK_1:10; set B1 = Ball (p1,r); set B2 = [#] (TOP-REAL n); reconsider W = [:(Ball (p1,r)),([#] (TOP-REAL n)):] as open Subset of [:(TOP-REAL n),(TOP-REAL n):] by BORSUK_1:6; take W ; ::_thesis: ( p in W & f1 .: W c= ].((f1 . p) - r),((f1 . p) + r).[ ) ( p1 in Ball (p1,r) & p2 in [#] (TOP-REAL n) ) by TOPGEN_5:13; hence p in W by A9, ZFMISC_1:def_2; ::_thesis: f1 .: W c= ].((f1 . p) - r),((f1 . p) + r).[ let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f1 .: W or y in ].((f1 . p) - r),((f1 . p) + r).[ ) assume y in f1 .: W ; ::_thesis: y in ].((f1 . p) - r),((f1 . p) + r).[ then consider x being Point of [:(TOP-REAL n),(TOP-REAL n):] such that A10: x in W and A11: f1 . x = y by FUNCT_2:65; consider x1, x2 being Point of (TOP-REAL n) such that A12: x = [x1,x2] by BORSUK_1:10; A13: ( f1 . (x1,x2) = x1 . i & f1 . (p1,p2) = p1 . i ) by A7; x1 in Ball (p1,r) by A10, A12, ZFMISC_1:87; then A14: |.(x1 - p1).| < r by TOPREAL9:7; dom (x1 - p1) = Seg n by FINSEQ_1:89; then (x1 . i) - (p1 . i) = (x1 - p1) . i by VALUED_1:13; then abs ((x1 . i) - (p1 . i)) <= |.(x1 - p1).| by A3, REAL_NS1:8; then abs ((f1 . x) - (f1 . p)) < r by A9, A12, A13, A14, XXREAL_0:2; hence y in ].((f1 . p) - r),((f1 . p) + r).[ by A11, RCOMP_1:1; ::_thesis: verum end; then A15: f1 is continuous by TOPS_4:21; for p being Point of [:(TOP-REAL n),(TOP-REAL n):] for r being positive real number ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f2 .: W c= ].((f2 . p) - r),((f2 . p) + r).[ ) proof let p be Point of [:(TOP-REAL n),(TOP-REAL n):]; ::_thesis: for r being positive real number ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f2 .: W c= ].((f2 . p) - r),((f2 . p) + r).[ ) let r be positive real number ; ::_thesis: ex W being open Subset of [:(TOP-REAL n),(TOP-REAL n):] st ( p in W & f2 .: W c= ].((f2 . p) - r),((f2 . p) + r).[ ) consider p1, p2 being Point of (TOP-REAL n) such that A16: p = [p1,p2] by BORSUK_1:10; set B1 = [#] (TOP-REAL n); set B2 = Ball (p2,r); reconsider W = [:([#] (TOP-REAL n)),(Ball (p2,r)):] as open Subset of [:(TOP-REAL n),(TOP-REAL n):] by BORSUK_1:6; take W ; ::_thesis: ( p in W & f2 .: W c= ].((f2 . p) - r),((f2 . p) + r).[ ) ( p1 in [#] (TOP-REAL n) & p2 in Ball (p2,r) ) by TOPGEN_5:13; hence p in W by A16, ZFMISC_1:def_2; ::_thesis: f2 .: W c= ].((f2 . p) - r),((f2 . p) + r).[ let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f2 .: W or y in ].((f2 . p) - r),((f2 . p) + r).[ ) assume y in f2 .: W ; ::_thesis: y in ].((f2 . p) - r),((f2 . p) + r).[ then consider x being Point of [:(TOP-REAL n),(TOP-REAL n):] such that A17: x in W and A18: f2 . x = y by FUNCT_2:65; consider x1, x2 being Point of (TOP-REAL n) such that A19: x = [x1,x2] by BORSUK_1:10; A20: ( f2 . (x1,x2) = x2 . i & f2 . (p1,p2) = p2 . i ) by A8; x2 in Ball (p2,r) by A17, A19, ZFMISC_1:87; then A21: |.(x2 - p2).| < r by TOPREAL9:7; dom (x2 - p2) = Seg n by FINSEQ_1:89; then (x2 . i) - (p2 . i) = (x2 - p2) . i by VALUED_1:13; then abs ((x2 . i) - (p2 . i)) <= |.(x2 - p2).| by A3, REAL_NS1:8; then abs ((f2 . x) - (f2 . p)) < r by A16, A19, A20, A21, XXREAL_0:2; hence y in ].((f2 . p) - r),((f2 . p) + r).[ by A18, RCOMP_1:1; ::_thesis: verum end; then f2 is continuous by TOPS_4:21; then consider g being Function of [:(TOP-REAL n),(TOP-REAL n):],R^1 such that A22: for p being Point of [:(TOP-REAL n),(TOP-REAL n):] for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 * r2 and A23: g is continuous by A15, JGRAPH_2:25; now__::_thesis:_for_a,_b_being_Point_of_(TOP-REAL_n)_holds_f_._(a,b)_=_g_._(a,b) let a, b be Point of (TOP-REAL n); ::_thesis: f . (a,b) = g . (a,b) A24: ( f1 . (a,b) = a . i & f2 . (a,b) = b . i ) by A7, A8; thus f . (a,b) = (a . i) * (b . i) by A6 .= g . [a,b] by A22, A24 .= g . (a,b) ; ::_thesis: verum end; hence (proj (((Seg n) --> R^1),i)) * F is continuous by A23, A4, A5, BINOP_1:2; ::_thesis: verum end; then F is continuous by WAYBEL18:6; hence TIMES n is continuous by A1, A2, YELLOW12:36; ::_thesis: verum end; suppose n is zero ; ::_thesis: TIMES n is continuous hence TIMES n is continuous by Th53; ::_thesis: verum end; end; end; end; theorem :: TOPREALC:59 for m, n being Nat for f being Function of (TOP-REAL m),(TOP-REAL n) st f is continuous holds f (-) is continuous Function of (TOP-REAL m),(TOP-REAL n) proof let m, n be Nat; ::_thesis: for f being Function of (TOP-REAL m),(TOP-REAL n) st f is continuous holds f (-) is continuous Function of (TOP-REAL m),(TOP-REAL n) let f be Function of (TOP-REAL m),(TOP-REAL n); ::_thesis: ( f is continuous implies f (-) is continuous Function of (TOP-REAL m),(TOP-REAL n) ) assume A1: f is continuous ; ::_thesis: f (-) is continuous Function of (TOP-REAL m),(TOP-REAL n) reconsider g = f (-) as Function of (TOP-REAL m),(TOP-REAL n) by Th34; for p being Point of (TOP-REAL m) for r being positive real number ex s being positive real number st g .: (Ball (p,s)) c= Ball ((g . p),r) proof let p be Point of (TOP-REAL m); ::_thesis: for r being positive real number ex s being positive real number st g .: (Ball (p,s)) c= Ball ((g . p),r) let r be positive real number ; ::_thesis: ex s being positive real number st g .: (Ball (p,s)) c= Ball ((g . p),r) consider s being positive real number such that A2: f .: (Ball ((- p),s)) c= Ball ((f . (- p)),r) by A1, TOPS_4:20; take s ; ::_thesis: g .: (Ball (p,s)) c= Ball ((g . p),r) let y be Element of (TOP-REAL n); :: according to LATTICE7:def_1 ::_thesis: ( not y in g .: (Ball (p,s)) or y in Ball ((g . p),r) ) assume y in g .: (Ball (p,s)) ; ::_thesis: y in Ball ((g . p),r) then consider x being Element of (TOP-REAL m) such that A3: x in Ball (p,s) and A4: g . x = y by FUNCT_2:65; dom g = the carrier of (TOP-REAL m) by FUNCT_2:def_1; then A5: ( g . x = f . (- x) & g . p = f . (- p) ) by VALUED_2:def_34; - x in Ball ((- p),s) by A3, Th23; then f . (- x) in f .: (Ball ((- p),s)) by FUNCT_2:35; hence y in Ball ((g . p),r) by A2, A4, A5; ::_thesis: verum end; hence f (-) is continuous Function of (TOP-REAL m),(TOP-REAL n) by TOPS_4:20; ::_thesis: verum end; registration let T be non empty TopSpace; let f be continuous Function of T,R^1; cluster - f -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = - f holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = - f implies F is continuous ) assume A1: F = - f ; ::_thesis: F is continuous consider g being Function of T,R^1 such that A2: for p being Point of T for r being real number st f . p = r holds g . p = - r and A3: g is continuous by JGRAPH_4:8; F = g proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = g . x thus F . x = - (f . x) by A1, VALUED_1:8 .= g . x by A2 ; ::_thesis: verum end; hence F is continuous by A3; ::_thesis: verum end; end; registration let T be non empty TopSpace; let f be non-empty continuous Function of T,R^1; clusterf " -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f " holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = f " implies F is continuous ) assume A1: F = f " ; ::_thesis: F is continuous dom f = the carrier of T by FUNCT_2:def_1; then for q being Point of T holds f . q <> 0 ; then consider g being Function of T,R^1 such that A2: for p being Point of T for r being real number st f . p = r holds g . p = 1 / r and A3: g is continuous by JGRAPH_2:26; F = g proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = g . x thus F . x = 1 / (f . x) by A1, VALUED_1:10 .= g . x by A2 ; ::_thesis: verum end; hence F is continuous by A3; ::_thesis: verum end; end; registration let T be non empty TopSpace; let f be continuous Function of T,R^1; let r be real number ; clusterr + f -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f + r holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = f + r implies F is continuous ) assume A1: F = f + r ; ::_thesis: F is continuous consider g being Function of T,R^1 such that A2: for p being Point of T for s being real number st f . p = s holds g . p = s + r and A3: g is continuous by JGRAPH_2:24; F = g proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = g . x thus F . x = (f . x) + r by A1, VALUED_1:2 .= g . x by A2 ; ::_thesis: verum end; hence F is continuous by A3; ::_thesis: verum end; clusterf - r -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f - r holds b1 is continuous ; clusterr (#) f -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f (#) r holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = f (#) r implies F is continuous ) assume A4: F = f (#) r ; ::_thesis: F is continuous consider g being Function of T,R^1 such that A5: for p being Point of T for s being real number st f . p = s holds g . p = r * s and A6: g is continuous by JGRAPH_2:23; F = g proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = g . x thus F . x = (f . x) * r by A4, VALUED_1:6 .= g . x by A5 ; ::_thesis: verum end; hence F is continuous by A6; ::_thesis: verum end; clusterf (/) r -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f (/) r holds b1 is continuous ; end; registration let T be non empty TopSpace; let f, g be continuous Function of T,R^1; clusterf + g -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f + g holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = f + g implies F is continuous ) assume A1: F = f + g ; ::_thesis: F is continuous consider h being Function of T,R^1 such that A2: for p being Point of T for r1, r2 being real number st f . p = r1 & g . p = r2 holds h . p = r1 + r2 and A3: h is continuous by JGRAPH_2:19; F = h proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = h . x thus F . x = (f . x) + (g . x) by A1, VALUED_1:1 .= h . x by A2 ; ::_thesis: verum end; hence F is continuous by A3; ::_thesis: verum end; clusterf - g -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f - g holds b1 is continuous proof f - g = f + (- g) ; hence for b1 being Function of T,R^1 st b1 = f - g holds b1 is continuous ; ::_thesis: verum end; clusterf (#) g -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f (#) g holds b1 is continuous proof let F be Function of T,R^1; ::_thesis: ( F = f (#) g implies F is continuous ) assume A4: F = f (#) g ; ::_thesis: F is continuous consider h being Function of T,R^1 such that A5: for p being Point of T for r1, r2 being real number st f . p = r1 & g . p = r2 holds h . p = r1 * r2 and A6: h is continuous by JGRAPH_2:25; F = h proof let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = h . x thus F . x = (f . x) * (g . x) by A4, VALUED_1:5 .= h . x by A5 ; ::_thesis: verum end; hence F is continuous by A6; ::_thesis: verum end; end; registration let T be non empty TopSpace; let f be continuous Function of T,R^1; let g be non-empty continuous Function of T,R^1; clusterf /" g -> continuous for Function of T,R^1; coherence for b1 being Function of T,R^1 st b1 = f /" g holds b1 is continuous proof f /" g = f (#) (g ") ; hence for b1 being Function of T,R^1 st b1 = f /" g holds b1 is continuous ; ::_thesis: verum end; end; registration let n be Nat; let T be non empty TopSpace; let f, g be continuous Function of T,(TOP-REAL n); clusterf <++> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <++> g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f <++> g implies h is continuous ) assume A1: h = f <++> g ; ::_thesis: h is continuous n in NAT by ORDINAL1:def_12; then consider F being Function of T,(TOP-REAL n) such that A2: for r being Point of T holds F . r = (f . r) + (g . r) and A3: F is continuous by JGRAPH_6:12; F = h proof A4: dom h = the carrier of T by FUNCT_2:def_1; let x be Point of T; :: according to FUNCT_2:def_8 ::_thesis: F . x = h . x thus F . x = (f . x) + (g . x) by A2 .= h . x by A1, A4, VALUED_2:def_45 ; ::_thesis: verum end; hence h is continuous by A3; ::_thesis: verum end; clusterf <--> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <--> g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f <--> g implies h is continuous ) assume A5: h = f <--> g ; ::_thesis: h is continuous A6: n in NAT by ORDINAL1:def_12; A7: for r being Point of T holds h . r = (f . r) - (g . r) proof let r be Point of T; ::_thesis: h . r = (f . r) - (g . r) dom h = the carrier of T by FUNCT_2:def_1; hence h . r = (f . r) - (g . r) by A5, VALUED_2:def_46; ::_thesis: verum end; for p being Point of T for V being Subset of (TOP-REAL n) st h . p in V & V is open holds ex W being Subset of T st ( p in W & W is open & h .: W c= V ) proof let p be Point of T; ::_thesis: for V being Subset of (TOP-REAL n) st h . p in V & V is open holds ex W being Subset of T st ( p in W & W is open & h .: W c= V ) let V be Subset of (TOP-REAL n); ::_thesis: ( h . p in V & V is open implies ex W being Subset of T st ( p in W & W is open & h .: W c= V ) ) assume ( h . p in V & V is open ) ; ::_thesis: ex W being Subset of T st ( p in W & W is open & h .: W c= V ) then A8: h . p in Int V by TOPS_1:23; reconsider r = h . p as Point of (Euclid n) by EUCLID:67; consider r0 being real number such that A9: ( r0 > 0 & Ball (r,r0) c= V ) by A8, GOBOARD6:5; reconsider r01 = f . p as Point of (Euclid n) by EUCLID:67; reconsider G1 = Ball (r01,(r0 / 2)) as Subset of (TOP-REAL n) by EUCLID:67; A10: f . p in G1 by A9, GOBOARD6:1; A11: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8; G1 is open by A11, TOPMETR:14, TOPS_3:76; then consider W1 being Subset of T such that A12: ( p in W1 & W1 is open & f .: W1 c= G1 ) by A10, JGRAPH_2:10; reconsider r02 = g . p as Point of (Euclid n) by EUCLID:67; reconsider G2 = Ball (r02,(r0 / 2)) as Subset of (TOP-REAL n) by EUCLID:67; A13: g . p in G2 by A9, GOBOARD6:1; G2 is open by A11, TOPMETR:14, TOPS_3:76; then consider W2 being Subset of T such that A14: ( p in W2 & W2 is open & g .: W2 c= G2 ) by A13, JGRAPH_2:10; take W = W1 /\ W2; ::_thesis: ( p in W & W is open & h .: W c= V ) thus p in W by A12, A14, XBOOLE_0:def_4; ::_thesis: ( W is open & h .: W c= V ) thus W is open by A12, A14; ::_thesis: h .: W c= V let x be Element of (TOP-REAL n); :: according to LATTICE7:def_1 ::_thesis: ( not x in h .: W or x in V ) assume x in h .: W ; ::_thesis: x in V then consider z being set such that A15: ( z in dom h & z in W & h . z = x ) by FUNCT_1:def_6; A16: z in W1 by A15, XBOOLE_0:def_4; reconsider pz = z as Point of T by A15; dom f = the carrier of T by FUNCT_2:def_1; then A17: f . pz in f .: W1 by A16, FUNCT_1:def_6; reconsider bb1 = f . pz as Point of (Euclid n) by EUCLID:67; dist (r01,bb1) < r0 / 2 by A12, A17, METRIC_1:11; then A18: |.((f . p) - (f . pz)).| < r0 / 2 by A6, JGRAPH_1:28; A19: z in W2 by A15, XBOOLE_0:def_4; dom g = the carrier of T by FUNCT_2:def_1; then A20: g . pz in g .: W2 by A19, FUNCT_1:def_6; reconsider bb2 = g . pz as Point of (Euclid n) by EUCLID:67; dist (r02,bb2) < r0 / 2 by A14, A20, METRIC_1:11; then A21: |.((g . p) - (g . pz)).| < r0 / 2 by A6, JGRAPH_1:28; A22: (f . pz) - (g . pz) = x by A7, A15; reconsider bb0 = (f . pz) - (g . pz) as Point of (Euclid n) by EUCLID:67; A23: h . pz = (f . pz) - (g . pz) by A7; ((f . p) - (g . p)) - ((f . pz) - (g . pz)) = (((f . p) - (g . p)) - (f . pz)) + (g . pz) by EUCLID:47 .= (((f . p) + (- (g . p))) + (- (f . pz))) + (g . pz) .= (((f . p) + (- (f . pz))) + (- (g . p))) + (g . pz) by EUCLID:26 .= (((f . p) - (f . pz)) - (g . p)) + (g . pz) .= (((f . p) - (f . pz)) + (g . pz)) + (- (g . p)) by EUCLID:26 .= ((f . p) - (f . pz)) + ((g . pz) - (g . p)) by EUCLID:26 ; then A24: |.(((f . p) - (g . p)) - ((f . pz) - (g . pz))).| <= |.((f . p) - (f . pz)).| + |.((g . pz) - (g . p)).| by A6, TOPRNS_1:29; A25: |.((g . p) - (g . pz)).| = |.(- ((g . pz) - (g . p))).| by EUCLID:44 .= |.((g . pz) - (g . p)).| by A6, TOPRNS_1:26 ; |.((f . p) - (f . pz)).| + |.((g . p) - (g . pz)).| < (r0 / 2) + (r0 / 2) by A18, A21, XREAL_1:8; then |.(((f . p) - (g . p)) - ((f . pz) - (g . pz))).| < r0 by A24, A25, XXREAL_0:2; then |.((h . p) - (h . pz)).| < r0 by A7, A23; then dist (r,bb0) < r0 by A6, A15, A22, JGRAPH_1:28; then x in Ball (r,r0) by A22, METRIC_1:11; hence x in V by A9; ::_thesis: verum end; hence h is continuous by JGRAPH_2:10; ::_thesis: verum end; clusterf <##> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <##> g holds b1 is continuous proof A26: f <##> g = (TIMES n) .: (f,g) by Th54; <:f,g:> is continuous Function of T,[:(TOP-REAL n),(TOP-REAL n):] by YELLOW12:41; hence for b1 being Function of T,(TOP-REAL n) st b1 = f <##> g holds b1 is continuous by A26; ::_thesis: verum end; end; registration let n be Nat; let T be non empty TopSpace; let f be continuous Function of T,(TOP-REAL n); let g be continuous Function of T,R^1; set I = incl (g,n); clusterf <+> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <+> g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f <+> g implies h is continuous ) assume A1: h = f <+> g ; ::_thesis: h is continuous reconsider G = f <++> (incl (g,n)) as Function of T,(TOP-REAL n) by Th39; h = G by A1, Th49; hence h is continuous ; ::_thesis: verum end; clusterf <-> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <-> g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f <-> g implies h is continuous ) assume A2: h = f <-> g ; ::_thesis: h is continuous reconsider G = f <--> (incl (g,n)) as Function of T,(TOP-REAL n) by Th40; h = G by A2, Th50; hence h is continuous ; ::_thesis: verum end; clusterf <#> g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f <#> g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f <#> g implies h is continuous ) assume A3: h = f <#> g ; ::_thesis: h is continuous reconsider G = f <##> (incl (g,n)) as Function of T,(TOP-REAL n) by Th41; h = G by A3, Th51; hence h is continuous ; ::_thesis: verum end; end; registration let n be Nat; let T be non empty TopSpace; let f be continuous Function of T,(TOP-REAL n); let g be non-empty continuous Function of T,R^1; clusterf g -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f g holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f g implies h is continuous ) reconsider g1 = g " as Function of T,R^1 ; g1 is continuous ; hence ( h = f g implies h is continuous ) ; ::_thesis: verum end; end; registration let n be Nat; let T be non empty TopSpace; let r be real number ; let f be continuous Function of T,(TOP-REAL n); clusterf [+] r -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f [+] r holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f [+] r implies h is continuous ) assume A1: h = f [+] r ; ::_thesis: h is continuous reconsider r = r as Element of R^1 by XREAL_0:def_1; dom f = the carrier of T by FUNCT_2:def_1; then h = f <+> (T --> r) by A1, Th29; hence h is continuous ; ::_thesis: verum end; clusterf [-] r -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f [-] r holds b1 is continuous ; clusterf [#] r -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f [#] r holds b1 is continuous proof let h be Function of T,(TOP-REAL n); ::_thesis: ( h = f [#] r implies h is continuous ) assume A2: h = f [#] r ; ::_thesis: h is continuous reconsider r = r as Element of R^1 by XREAL_0:def_1; dom f = the carrier of T by FUNCT_2:def_1; then h = f <#> (T --> r) by A2, Th31; hence h is continuous ; ::_thesis: verum end; clusterf [/] r -> continuous for Function of T,(TOP-REAL n); coherence for b1 being Function of T,(TOP-REAL n) st b1 = f [/] r holds b1 is continuous ; end; theorem Th60: :: TOPREALC:60 for r being non negative real number for n being non zero Nat for p being Point of (Tcircle ((0. (TOP-REAL n)),r)) holds - p is Point of (Tcircle ((0. (TOP-REAL n)),r)) proof let r be non negative real number ; ::_thesis: for n being non zero Nat for p being Point of (Tcircle ((0. (TOP-REAL n)),r)) holds - p is Point of (Tcircle ((0. (TOP-REAL n)),r)) let n be non zero Nat; ::_thesis: for p being Point of (Tcircle ((0. (TOP-REAL n)),r)) holds - p is Point of (Tcircle ((0. (TOP-REAL n)),r)) let p be Point of (Tcircle ((0. (TOP-REAL n)),r)); ::_thesis: - p is Point of (Tcircle ((0. (TOP-REAL n)),r)) reconsider p1 = p as Point of (TOP-REAL n) by PRE_TOPC:25; A1: n in NAT by ORDINAL1:def_12; then A2: the carrier of (Tcircle ((0. (TOP-REAL n)),r)) = Sphere ((0. (TOP-REAL n)),r) by TOPREALB:9; |.((- p1) - (0. (TOP-REAL n))).| = |.(- p1).| by RLVECT_1:13 .= |.p1.| by EUCLID:71 .= |.(p1 - (0. (TOP-REAL n))).| by RLVECT_1:13 .= r by A1, A2, TOPREAL9:9 ; hence - p is Point of (Tcircle ((0. (TOP-REAL n)),r)) by A2, A1, TOPREAL9:9; ::_thesis: verum end; theorem Th61: :: TOPREALC:61 for n being Nat for r being non negative real number for f being Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) holds f (-) is Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) proof let n be Nat; ::_thesis: for r being non negative real number for f being Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) holds f (-) is Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) let r be non negative real number ; ::_thesis: for f being Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) holds f (-) is Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) set X = the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)); let f be Function of the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n); ::_thesis: f (-) is Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) set g = f (-) ; A1: dom (f (-)) = dom f by VALUED_2:def_34; A2: dom f = the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) by FUNCT_2:def_1; for x being set st x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) holds (f (-)) . x in the carrier of (TOP-REAL n) proof let x be set ; ::_thesis: ( x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) implies (f (-)) . x in the carrier of (TOP-REAL n) ) assume x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) ; ::_thesis: (f (-)) . x in the carrier of (TOP-REAL n) then reconsider x = x as Element of the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) ; reconsider y = - x as Element of the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) by Th60; reconsider f = f as Function of the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) ; (f (-)) . x = f . y by A1, A2, VALUED_2:def_34; hence (f (-)) . x in the carrier of (TOP-REAL n) ; ::_thesis: verum end; hence f (-) is Function of (Tcircle ((0. (TOP-REAL (n + 1))),r)),(TOP-REAL n) by A1, A2, FUNCT_2:3; ::_thesis: verum end; definition let n be Nat; let r be non negative real number ; let X be Subset of (Tcircle ((0. (TOP-REAL (n + 1))),r)); :: original: (-) redefine func (-) X -> Subset of (Tcircle ((0. (TOP-REAL (n + 1))),r)); coherence (-) X is Subset of (Tcircle ((0. (TOP-REAL (n + 1))),r)) proof set T = Tcircle ((0. (TOP-REAL (n + 1))),r); (-) X c= the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (-) X or x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) ) assume A1: x in (-) X ; ::_thesis: x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) then reconsider x = x as Element of (-) X ; - x in X by A1, Th24; then - (- x) is Point of (Tcircle ((0. (TOP-REAL (n + 1))),r)) by Th60; hence x in the carrier of (Tcircle ((0. (TOP-REAL (n + 1))),r)) ; ::_thesis: verum end; hence (-) X is Subset of (Tcircle ((0. (TOP-REAL (n + 1))),r)) ; ::_thesis: verum end; end; registration let m be Nat; let r be non negative real number ; let X be open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)); cluster (-) X -> open for Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)); coherence for b1 being Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) st b1 = (-) X holds b1 is open proof set T = Tcircle ((0. (TOP-REAL (m + 1))),r); ex G being Subset of (TOP-REAL (m + 1)) st ( G is open & (-) X = G /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) ) proof consider G being Subset of (TOP-REAL (m + 1)) such that A1: G is open and A2: X = G /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by TSP_1:def_1; take (-) G ; ::_thesis: ( (-) G is open & (-) X = ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) ) thus (-) G is open by A1; ::_thesis: (-) X = ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) thus (-) X c= ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) :: according to XBOOLE_0:def_10 ::_thesis: ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) c= (-) X proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (-) X or x in ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) ) assume A3: x in (-) X ; ::_thesis: x in ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) then reconsider x = x as Element of (-) X ; - x in X by A3, Th24; then - x in G by A2, XBOOLE_0:def_4; then x in (-) G by Th24; hence x in ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by A3, XBOOLE_0:def_4; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) or x in (-) X ) assume A4: x in ((-) G) /\ the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) ; ::_thesis: x in (-) X then reconsider x = x as Element of (-) G by XBOOLE_0:def_4; x in (-) G by A4, XBOOLE_0:def_4; then A5: - x in G by Th24; x in the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by A4, XBOOLE_0:def_4; then - x is Point of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by Th60; then - x in X by A2, A5, XBOOLE_0:def_4; hence x in (-) X by Th24; ::_thesis: verum end; hence for b1 being Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) st b1 = (-) X holds b1 is open by TSP_1:def_1; ::_thesis: verum end; end; theorem :: TOPREALC:62 for m being Nat for r being non negative real number for f being continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) holds f (-) is continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) proof let m be Nat; ::_thesis: for r being non negative real number for f being continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) holds f (-) is continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) let r be non negative real number ; ::_thesis: for f being continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) holds f (-) is continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) set T = Tcircle ((0. (TOP-REAL (m + 1))),r); let f be continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m); ::_thesis: f (-) is continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) reconsider g = f (-) as Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) by Th61; for p being Point of (Tcircle ((0. (TOP-REAL (m + 1))),r)) for r being positive real number ex W being open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) st ( p in W & g .: W c= Ball ((g . p),r) ) proof let p be Point of (Tcircle ((0. (TOP-REAL (m + 1))),r)); ::_thesis: for r being positive real number ex W being open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) st ( p in W & g .: W c= Ball ((g . p),r) ) let r be positive real number ; ::_thesis: ex W being open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) st ( p in W & g .: W c= Ball ((g . p),r) ) reconsider q = - p as Point of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by Th60; consider W being open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) such that A1: q in W and A2: f .: W c= Ball ((f . q),r) by TOPS_4:18; reconsider W1 = (-) W as open Subset of (Tcircle ((0. (TOP-REAL (m + 1))),r)) ; take W1 ; ::_thesis: ( p in W1 & g .: W1 c= Ball ((g . p),r) ) - q in W1 by A1, Def3; hence p in W1 ; ::_thesis: g .: W1 c= Ball ((g . p),r) let y be Element of (TOP-REAL m); :: according to LATTICE7:def_1 ::_thesis: ( not y in g .: W1 or y in Ball ((g . p),r) ) assume y in g .: W1 ; ::_thesis: y in Ball ((g . p),r) then consider x being Element of (Tcircle ((0. (TOP-REAL (m + 1))),r)) such that A3: x in W1 and A4: g . x = y by FUNCT_2:65; dom g = the carrier of (Tcircle ((0. (TOP-REAL (m + 1))),r)) by FUNCT_2:def_1; then A5: ( g . x = f . (- x) & g . p = f . (- p) ) by VALUED_2:def_34; - x in (-) W1 by A3, Def3; then f . (- x) in f .: W by FUNCT_2:35; hence y in Ball ((g . p),r) by A2, A4, A5; ::_thesis: verum end; hence f (-) is continuous Function of (Tcircle ((0. (TOP-REAL (m + 1))),r)),(TOP-REAL m) by TOPS_4:18; ::_thesis: verum end;
distilabel_metadata.role: user