for a, b being Real
for f being PartFunc of REAL,REAL holds
( f is_strictly_convex_on [.a,b.] iff ( [.a,b.] c= dom f & ( for p being Real st 0 < p & p < 1 holds
for r, s being Real st r in [.a,b.] & s in [.a,b.] & r <> s holds
f . ((p * r) + ((1 - p) * s)) < (p * (f . r)) + ((1 - p) * (f . s)) ) ) )