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 object st y in Z holds
y in dom f by A1, FUNCT_1:11;
then A3: Z c= dom f by TARSKI:def 3;
then A4: f is_differentiable_on Z by A2, FDIFF_1:23;
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_1:9;
sin is_differentiable_in f . x by SIN_COS:64;
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_1:9;
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)) )
A8: sin is_differentiable_in f . x by SIN_COS:64;
assume A9: x in Z ; :: thesis: ((sin * f) `| Z) . x = a * (cos . ((a * x) + b))
then f is_differentiable_in x by A4, FDIFF_1:9;
then diff ((sin * f),x) = (diff (sin,(f . x))) * (diff (f,x)) by A8, FDIFF_2:13
.= (cos . (f . x)) * (diff (f,x)) by SIN_COS:64
.= (cos . ((a * x) + b)) * (diff (f,x)) by A2, A9
.= (cos . ((a * x) + b)) * ((f `| Z) . x) by A4, A9, FDIFF_1:def 7
.= a * (cos . ((a * x) + b)) by A2, A3, A9, FDIFF_1:23 ;
hence ((sin * f) `| Z) . x = a * (cos . ((a * x) + b)) by A7, A9, FDIFF_1:def 7; :: 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_1:9; :: thesis: verum