Library GeoCoq.Elements.OriginalProofs.lemma_equalangleshelper

Require Export GeoCoq.Elements.OriginalProofs.lemma_ray3.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_equalangleshelper :
   ∀ A B C a b c p q,
   CongA A B C a b c → Out b a p → Out b c q →
   CongA A B C p b q.
Proof.
intros.
let Tf:=fresh in
assert (Tf:∃ U V u v, (Out B A U ∧ Out B C V ∧ Out b a u ∧ Out b c v ∧ Cong B U b u ∧ Cong B V b v ∧ Cong U V u v ∧ nCol A B C)) by (conclude_def CongA );destruct Tf as [U[V[u[v]]]];spliter.
assert (Out b p u) by (conclude lemma_ray3).
assert (Out b q v) by (conclude lemma_ray3).
assert (CongA A B C p b q) by (conclude_def CongA ).
close.
Qed.

End Euclid.