Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthancongruence.

Section Euclid.

Context `{Ax:euclidean_neutral_ruler_compass}.

Lemma proposition_03 : 
   forall A B C D E F, 
   Lt C D A B -> Cong E F A B ->
   exists X, BetS E X F /\ Cong E X C D.
Proof.
(* Goal: forall (A B C D E F : @Point Ax0) (_ : @Lt Ax0 C D A B) (_ : @Cong Ax0 E F A B), @ex (@Point Ax0) (fun X : @Point Ax0 => and (@BetS Ax0 E X F) (@Cong Ax0 E X C D)) *)
intros.
(* Goal: @ex (@Point Ax0) (fun X : @Point Ax0 => and (@BetS Ax0 E X F) (@Cong Ax0 E X C D)) *)
assert (Cong A B E F) by (conclude lemma_congruencesymmetric).
(* Goal: @ex (@Point Ax0) (fun X : @Point Ax0 => and (@BetS Ax0 E X F) (@Cong Ax0 E X C D)) *)
assert (Lt C D E F) by (conclude lemma_lessthancongruence).
let Tf:=fresh in
assert (Tf:exists G, (BetS E G F /\ Cong E G C D)) by (conclude_def Lt );destruct Tf as [G];spliter.
(* Goal: @ex (@Point Ax0) (fun X : @Point Ax0 => and (@BetS Ax0 E X F) (@Cong Ax0 E X C D)) *)
close.
Qed.

End Euclid.


