let a, b be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (cos * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( cos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom (cos * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( cos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (cos * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) implies ( cos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) ) )

assume that
A1: Z c= dom (cos * f) and
A2: for x being Real st x in Z holds
f . x = (a * x) + b ; :: thesis: ( cos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) )

for y being object st y in Z holds
y in dom f by ;
then A3: Z c= dom f by TARSKI:def 3;
then A4: f is_differentiable_on Z by ;
A5: for x being Real st x in Z holds
cos * f is_differentiable_in x
proof end;
then A7: cos * f is_differentiable_on Z by ;
for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b)))
proof
let x be Real; :: thesis: ( x in Z implies ((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) )
A8: diff (cos,(f . x)) = - (sin . (f . x)) by SIN_COS:63;
A9: cos is_differentiable_in f . x by SIN_COS:63;
assume A10: x in Z ; :: thesis: ((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b)))
then f is_differentiable_in x by ;
then diff ((cos * f),x) = (diff (cos,(f . x))) * (diff (f,x)) by
.= - ((sin . (f . x)) * (diff (f,x))) by A8
.= - ((sin . ((a * x) + b)) * (diff (f,x))) by
.= - ((sin . ((a * x) + b)) * ((f `| Z) . x)) by
.= - (a * (sin . ((a * x) + b))) by ;
hence ((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) by ; :: thesis: verum
end;
hence ( cos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) ) by ; :: thesis: verum