# Library GeoCoq.Elements.OriginalProofs.proposition_03

Require Export GeoCoq.Elements.OriginalProofs.lemma_lessthancongruence.

Section Euclid.

Context `{Ax:euclidean_neutral_ruler_compass}.

Lemma proposition_03 :

∀ A B C D E F,

neq A B → neq C D → Lt C D A B → Cong E F A B →

∃ X, BetS E X F ∧ Cong E X C D.

Proof.

intros.

assert (Cong A B E F) by (conclude lemma_congruencesymmetric).

assert (Lt C D E F) by (conclude lemma_lessthancongruence).

let Tf:=fresh in

assert (Tf:∃ G, (BetS E G F ∧ Cong E G C D)) by (conclude_def Lt );destruct Tf as [G];spliter.

close.

Qed.

End Euclid.

Section Euclid.

Context `{Ax:euclidean_neutral_ruler_compass}.

Lemma proposition_03 :

∀ A B C D E F,

neq A B → neq C D → Lt C D A B → Cong E F A B →

∃ X, BetS E X F ∧ Cong E X C D.

Proof.

intros.

assert (Cong A B E F) by (conclude lemma_congruencesymmetric).

assert (Lt C D E F) by (conclude lemma_lessthancongruence).

let Tf:=fresh in

assert (Tf:∃ G, (BetS E G F ∧ Cong E G C D)) by (conclude_def Lt );destruct Tf as [G];spliter.

close.

Qed.

End Euclid.