Definition contraction a b k := k *: (b - a) + a.

Lemma betSP' a b c (r := betR a b c) :
reflect ([ /\ b 0 0 - a 0 0 = r * (c 0 0 - a 0 0),
b 0 1 - a 0 1 = r * (c 0 1 - a 0 1), 0 < r & r < 1])
(betS a b c).