text: let K be Field; :: thesis: for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 holds
  - (M1 + M2) = (- M1) + (- M2)
  
  let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 implies - (M1 + M2) = (- M1) + (- M2) )
  assume A1: ( len M1 = len M2 & width M1 = width M2 ) ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
  A2: width (- M1) = width M1 by MATRIX_3:def 2;
  then A3: width ((- M1) + (- M2)) = width M1 by MATRIX_3:def 3;
  A4: ( len (M1 + M2) = len M1 & width (M1 + M2) = width M1 ) by MATRIX_3:def 3;
  A5: len (- M1) = len M1 by MATRIX_3:def 2;
  then A6: len ((- M1) + (- M2)) = len M1 by MATRIX_3:def 3;
  A7: ( len (- M2) = len M2 & width (- M2) = width M2 ) by MATRIX_3:def 2;
  per cases ( len M1 > 0 or len M1 = 0 ) by NAT_1:3;
  suppose A8: len M1 > 0 ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
  then A9: M2 is Matrix of len M1, width M1,K by ;
  A10: M1 is Matrix of len M1, width M1,K by ;
  (M1 + M2) + ((- M1) + (- M2)) = (M1 + M2) + ((- M2) + (- M1)) by
  .= ((M1 + M2) + (- M2)) + (- M1) by
  .= (M1 + (M2 + (- M2))) + (- M1) by
  .= (M1 + (0. (K,(len M1),(width M1)))) + (- M1) by
  .= M1 + (- M1) by
  .= 0. (K,(len M1),(width M1)) by ;
  hence - (M1 + M2) = (- M1) + (- M2) by A4, A6, A3, Th8; :: thesis: verum
  end;
  suppose A11: len M1 = 0 ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
  then len (- M1) = 0 by MATRIX_3:def 2;
  then A12: len ((- M1) + (- M2)) = 0 by MATRIX_3:def 3;
  len (M1 + M2) = 0 by ;
  then len (- (M1 + M2)) = 0 by MATRIX_3:def 2;
  hence - (M1 + M2) = (- M1) + (- M2) by ; :: thesis: verum
  end;
  end;