let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom (exp_R (#) f) & ( for x being Real st x in Z holds
f . x = x - 1 ) holds
( exp_R (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) f) `| Z) . x = x * (exp_R . x) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (exp_R (#) f) & ( for x being Real st x in Z holds
f . x = x - 1 ) implies ( exp_R (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) f) `| Z) . x = x * (exp_R . x) ) ) )
assume that
A1: Z c= dom (exp_R (#) f) and
A2: for x being Real st x in Z holds
f . x = x - 1 ; :: thesis: ( exp_R (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) f) `| Z) . x = x * (exp_R . x) ) )
Z c= (dom f) /\ (dom exp_R ) by A1, VALUED_1:def 4;
then A3: Z c= dom f by XBOOLE_1:18;
A4: for x being Real st x in Z holds
f . x = (1 * x) + (- 1)

proof

let x be Real; :: thesis: ( x in Z implies f . x = (1 * x) + (- 1) )
assume A5: x in Z ; :: thesis: f . x = (1 * x) + (- 1)
(1 * x) + (- 1) = (1 * x) - 1 ;
hence f . x = (1 * x) + (- 1) by A2, A5; :: thesis: verum

end;

then A6: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 1 ) ) by A3, FDIFF_1:31;
A7: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;

now

let x be Real; :: thesis: ( x in Z implies ((exp_R (#) f) `| Z) . x = x * (exp_R . x) )
assume A8: x in Z ; :: thesis: ((exp_R (#) f) `| Z) . x = x * (exp_R . x)
hence ((exp_R (#) f) `| Z) . x = ((f . x) * (diff exp_R ,x)) + ((exp_R . x) * (diff f,x)) by A1, A6, A7, FDIFF_1:29
.= ((x - 1) * (diff exp_R ,x)) + ((exp_R . x) * (diff f,x)) by A2, A8
.= ((x - 1) * (exp_R . x)) + ((exp_R . x) * (diff f,x)) by TAYLOR_1:16
.= ((x - 1) * (exp_R . x)) + ((exp_R . x) * ((f `| Z) . x)) by A6, A8, FDIFF_1:def 8
.= ((x - 1) * (exp_R . x)) + ((exp_R . x) * 1) by A3, A4, A8, FDIFF_1:31
.= x * (exp_R . x) ;
:: thesis: verum

end;

hence ( exp_R (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) f) `| Z) . x = x * (exp_R . x) ) ) by A1, A6, A7, FDIFF_1:29; :: thesis: verum