# Library GeoCoq.Elements.OriginalProofs.lemma_3_6a

Require Export GeoCoq.Elements.OriginalProofs.euclidean_tactics.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_3_6a :

∀ A B C D,

BetS A B C → BetS A C D →

BetS B C D.

Proof.

intros.

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

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

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

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

close.

Qed.

End Euclid.

Section Euclid.

Context `{Ax:euclidean_neutral}.

Lemma lemma_3_6a :

∀ A B C D,

BetS A B C → BetS A C D →

BetS B C D.

Proof.

intros.

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

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

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

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

close.

Qed.

End Euclid.