# Library GeoCoq.Elements.OriginalProofs.lemma_3_7b

Require Export GeoCoq.Elements.OriginalProofs.lemma_3_7a.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_3_7b :

∀ A B C D,

BetS A B C → BetS B C D →

BetS A B D.

Proof.

intros.

assert (BetS C B A) by (conclude axiom_betweennesssymmetry).

assert (BetS D C B) by (conclude axiom_betweennesssymmetry).

assert (BetS D B A) by (conclude lemma_3_7a).

assert (BetS A B D) by (conclude axiom_betweennesssymmetry).

close.

Qed.

End Euclid.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_3_7b :

∀ A B C D,

BetS A B C → BetS B C D →

BetS A B D.

Proof.

intros.

assert (BetS C B A) by (conclude axiom_betweennesssymmetry).

assert (BetS D C B) by (conclude axiom_betweennesssymmetry).

assert (BetS D B A) by (conclude lemma_3_7a).

assert (BetS A B D) by (conclude axiom_betweennesssymmetry).

close.

Qed.

End Euclid.