let a, b be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom (sin * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( sin * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * f) `| Z) . x = a * (cos . ((a * x) + b)) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom (sin * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( sin * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * f) `| Z) . x = a * (cos . ((a * x) + b)) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (sin * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) implies ( sin * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * f) `| Z) . x = a * (cos . ((a * x) + b)) ) ) )
assume that
A1: Z c= dom (sin * f) and
A2: for x being Real st x in Z holds
f . x = (a * x) + b ; :: thesis: ( sin * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * f) `| Z) . x = a * (cos . ((a * x) + b)) ) )
for y being set st y in Z holds
y in dom f by A1, FUNCT_1:21;
then A3: Z c= dom f by TARSKI:def 3;
then A4: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = a ) ) by A2, FDIFF_1:31;
A5: for x being Real st x in Z holds
sin * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sin * f is_differentiable_in x )
assume x in Z ; :: thesis: sin * f is_differentiable_in x
then A6: f is_differentiable_in x by A4, FDIFF_1:16;
sin is_differentiable_in f . x by SIN_COS:69;
hence sin * f is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

end;

then A7: sin * f is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((sin * f) `| Z) . x = a * (cos . ((a * x) + b))

proof

let x be Real; :: thesis: ( x in Z implies ((sin * f) `| Z) . x = a * (cos . ((a * x) + b)) )
assume A8: x in Z ; :: thesis: ((sin * f) `| Z) . x = a * (cos . ((a * x) + b))
then A9: f is_differentiable_in x by A4, FDIFF_1:16;
sin is_differentiable_in f . x by SIN_COS:69;
then diff (sin * f),x = (diff sin ,(f . x)) * (diff f,x) by A9, FDIFF_2:13
.= (cos . (f . x)) * (diff f,x) by SIN_COS:69
.= (cos . ((a * x) + b)) * (diff f,x) by A2, A8
.= (cos . ((a * x) + b)) * ((f `| Z) . x) by A4, A8, FDIFF_1:def 8
.= a * (cos . ((a * x) + b)) by A2, A3, A8, FDIFF_1:31 ;
hence ((sin * f) `| Z) . x = a * (cos . ((a * x) + b)) by A7, A8, FDIFF_1:def 8; :: thesis: verum

end;

hence ( sin * f is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * f) `| Z) . x = a * (cos . ((a * x) + b)) ) ) by A1, A5, FDIFF_1:16; :: thesis: verum