Section between_identity_independent.
Inductive Point :=
P0 | P1 | P2 | P3.
Definition Bet (A B C : Point) :=
A = B \/ B = C \/ A = C.
Definition Cong (A B C D : Point) :=
(A = B /\ C = D) \/ (A <> B /\ C <> D).

Lemma euclid : forall A B C,
~ (Bet A B C \/ Bet B C A \/ Bet C A B) -> exists CC, Cong A CC B CC /\ Cong A CC C CC.