Section TarskiGe1.
Variable R : realFieldType.
Variable n : nat.
Implicit Types (a b c d : (@ Vector R n)).
Definition cong a b c d := (b - a) *m (b - a)^T == (d - c) *m (d - c)^T.
Definition bet a b c := betE a b c || betS a b c.
Definition extension a b k := k^-1 *: (b - a) + a.
Definition contraction a b k := k *: (b - a) + a.

Lemma euclid'_aux a b c d k1 (k2 := betR b d c) :
0 < k1 -> k1 < 1 -> bet b d c -> b != c ->
bet (extension a b k1) (extension a d k1) (extension a c k1).