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 ) holds
( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * f) `| Z) . 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 ) implies ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * f) `| Z) . 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 ; :: thesis: ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * f) `| Z) . x = - (exp_R (- x)) ) )
A3: for x being Real st x in Z holds
f . x = ((- 1) * x) + 0

proof

let x be Real; :: thesis: ( x in Z implies f . x = ((- 1) * x) + 0 )
assume x in Z ; :: thesis: f . x = ((- 1) * x) + 0
then f . x = - x by A2
.= ((- 1) * x) + 0 ;
hence f . x = ((- 1) * x) + 0 ; :: thesis: verum

end;

for y being object st y in Z holds
y in dom f by A1, FUNCT_1:11;
then A4: Z c= dom f by TARSKI:def 3;
then A5: f is_differentiable_on Z by A3, FDIFF_1:23;
A6: for x being Real st x in Z holds
exp_R * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies exp_R * f is_differentiable_in x )
assume x in Z ; :: thesis: exp_R * f is_differentiable_in x
then f is_differentiable_in x by A5, FDIFF_1:9;
hence exp_R * f is_differentiable_in x by TAYLOR_1:19; :: thesis: verum

end;

then A7: exp_R * f is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((exp_R * f) `| Z) . x = - (exp_R (- x))

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R * f) `| Z) . x = - (exp_R (- x)) )
assume A8: x in Z ; :: thesis: ((exp_R * f) `| Z) . x = - (exp_R (- x))
then f is_differentiable_in x by A5, FDIFF_1:9;
then diff ((exp_R * f),x) = (exp_R . (f . x)) * (diff (f,x)) by TAYLOR_1:19
.= (exp_R . (f . x)) * ((f `| Z) . x) by A5, A8, FDIFF_1:def 7
.= (exp_R . (f . x)) * (- 1) by A4, A3, A8, FDIFF_1:23
.= (exp_R . (- x)) * (- 1) by A2, A8
.= - (exp_R . (- x))
.= - (exp_R (- x)) by SIN_COS:def 23 ;
hence ((exp_R * f) `| Z) . x = - (exp_R (- x)) by A7, A8, FDIFF_1:def 7; :: 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 = - (exp_R (- x)) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum