let p be Real; :: thesis: for f being PartFunc of REAL,REAL st f = compreal holds
( (1 / 2) (#) (exp_R - (exp_R * f)) is_differentiable_in p & diff (((1 / 2) (#) (exp_R - (exp_R * f))),p) = (1 / 2) * (diff ((exp_R - (exp_R * f)),p)) )
let f be PartFunc of REAL,REAL; :: thesis: ( f = compreal implies ( (1 / 2) (#) (exp_R - (exp_R * f)) is_differentiable_in p & diff (((1 / 2) (#) (exp_R - (exp_R * f))),p) = (1 / 2) * (diff ((exp_R - (exp_R * f)),p)) ) )
assume f = compreal ; :: thesis: ( (1 / 2) (#) (exp_R - (exp_R * f)) is_differentiable_in p & diff (((1 / 2) (#) (exp_R - (exp_R * f))),p) = (1 / 2) * (diff ((exp_R - (exp_R * f)),p)) )
then ( p is Real & exp_R - (exp_R * f) is_differentiable_in p ) by Lm15;
hence ( (1 / 2) (#) (exp_R - (exp_R * f)) is_differentiable_in p & diff (((1 / 2) (#) (exp_R - (exp_R * f))),p) = (1 / 2) * (diff ((exp_R - (exp_R * f)),p)) ) by FDIFF_1:15; :: thesis: verum