Require Export GeoCoq.Elements.OriginalProofs.proposition_06a.
Require Export GeoCoq.Elements.OriginalProofs.lemma_trichotomy1.

Section Euclid.

Context `{Ax:euclidean_neutral_ruler_compass}.

Lemma proposition_06 : 
   forall A B C, 
   Triangle A B C -> CongA A B C A C B ->
   Cong A B A C.
Proof.
(* Goal: forall (A B C : @Point Ax0) (_ : @Triangle Ax0 A B C) (_ : @CongA Ax0 A B C A C B), @Cong Ax0 A B A C *)
intros.
(* Goal: @Cong Ax0 A B A C *)
assert (~ Lt A C A B) by (conclude proposition_06a).
(* Goal: @Cong Ax0 A B A C *)
assert (nCol A B C) by (conclude_def Triangle ).
(* Goal: @Cong Ax0 A B A C *)
assert (~ Col A C B).
(* Goal: not (@Col Ax0 A C B) *)
(* Goal: @Cong Ax0 A B A C *)
 {
(* Goal: not (@Col Ax0 A C B) *)
 intro.
(* Goal: False *)
 assert (Col A B C) by (forward_using lemma_collinearorder).
(* Goal: False *)
 contradict.
 }
(* Goal: @Cong Ax0 A B A C *)
assert (Triangle A C B) by (conclude_def Triangle ).
(* Goal: @Cong Ax0 A B A C *)
assert (CongA A C B A B C) by (conclude lemma_equalanglessymmetric).
(* Goal: @Cong Ax0 A B A C *)
assert (~ Lt A B A C) by (conclude proposition_06a).
(* Goal: @Cong Ax0 A B A C *)
assert (neq A B) by (forward_using lemma_angledistinct).
(* Goal: @Cong Ax0 A B A C *)
assert (neq A C) by (forward_using lemma_angledistinct).
(* Goal: @Cong Ax0 A B A C *)
assert (Cong A B A C) by (conclude lemma_trichotomy1).
(* Goal: @Cong Ax0 A B A C *)
close.
Qed.

End Euclid.


