let p, x0 be Real; :: thesis: for f being PartFunc of REAL,REAL st f is_differentiable_in x0 & f . x0 > 0 holds
( (#R p) * f is_differentiable_in x0 & diff (((#R p) * f),x0) = (p * ((f . x0) #R (p - 1))) * (diff (f,x0)) )
let f be PartFunc of REAL,REAL; :: thesis: ( f is_differentiable_in x0 & f . x0 > 0 implies ( (#R p) * f is_differentiable_in x0 & diff (((#R p) * f),x0) = (p * ((f . x0) #R (p - 1))) * (diff (f,x0)) ) )
assume that
A1: f is_differentiable_in x0 and
A2: f . x0 > 0 ; :: thesis: ( (#R p) * f is_differentiable_in x0 & diff (((#R p) * f),x0) = (p * ((f . x0) #R (p - 1))) * (diff (f,x0)) )
A3: #R p is_differentiable_in f . x0 by A2, Th21;
hence (#R p) * f is_differentiable_in x0 by A1, FDIFF_2:13; :: thesis: diff (((#R p) * f),x0) = (p * ((f . x0) #R (p - 1))) * (diff (f,x0))
thus diff (((#R p) * f),x0) = (diff ((#R p),(f . x0))) * (diff (f,x0)) by A1, A3, FDIFF_2:13
.= (p * ((f . x0) #R (p - 1))) * (diff (f,x0)) by A2, Th21 ; :: thesis: verum