# Library GeoCoq.Elements.OriginalProofs.lemma_RTcongruence

Require Export GeoCoq.Elements.OriginalProofs.lemma_equalanglestransitive.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_RTcongruence :

∀ A B C D E F P Q R,

RT A B C D E F → CongA A B C P Q R →

RT P Q R D E F.

Proof.

intros.

let Tf:=fresh in

assert (Tf:∃ a b c d e, (Supp a b c d e ∧ CongA A B C a b c ∧ CongA D E F d b e)) by (conclude_def RT );destruct Tf as [a[b[c[d[e]]]]];spliter.

assert (CongA P Q R A B C) by (conclude lemma_equalanglessymmetric).

assert (CongA P Q R a b c) by (conclude lemma_equalanglestransitive).

assert (RT P Q R D E F) by (conclude_def RT ).

close.

Qed.

End Euclid.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_RTcongruence :

∀ A B C D E F P Q R,

RT A B C D E F → CongA A B C P Q R →

RT P Q R D E F.

Proof.

intros.

let Tf:=fresh in

assert (Tf:∃ a b c d e, (Supp a b c d e ∧ CongA A B C a b c ∧ CongA D E F d b e)) by (conclude_def RT );destruct Tf as [a[b[c[d[e]]]]];spliter.

assert (CongA P Q R A B C) by (conclude lemma_equalanglessymmetric).

assert (CongA P Q R a b c) by (conclude lemma_equalanglestransitive).

assert (RT P Q R D E F) by (conclude_def RT ).

close.

Qed.

End Euclid.