Definition bet a b c := betE a b c || betS a b c.

Lemma euclid a b c d t (k1 := betR a d t) (k2 := betR b d c) : bet a d t -> bet b d c -> b <> d -> d <> c -> ~ (bet a b c \/ bet b c a \/ bet c a b) -> exists x y, bet a b x /\ bet a c y /\ bet x t y.