theorem :: ROLLE:4

for x, t being Real st 0 < t holds
for f being PartFunc of REAL,REAL st [.x,(x + t).] c= dom f & f | [.x,(x + t).] is continuous & f is_differentiable_on ].x,(x + t).[ holds
ex s being Real st
( 0 < s & s < 1 & f . (x + t) = (f . x) + (t * (diff (f,(x + (s * t))))) )