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 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

cos * f is_differentiable_in x

for x being Real st x in Z holds

((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b)))

((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) ) by A1, A5, FDIFF_1:9; :: thesis: verum

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 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

cos * f is_differentiable_in x

proof

then A7:
cos * f is_differentiable_on Z
by A1, FDIFF_1:9;
let x be Real; :: thesis: ( x in Z implies cos * f is_differentiable_in x )

assume x in Z ; :: thesis: cos * f is_differentiable_in x

then A6: f is_differentiable_in x by A4, FDIFF_1:9;

cos is_differentiable_in f . x by SIN_COS:63;

hence cos * f is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

end;assume x in Z ; :: thesis: cos * f is_differentiable_in x

then A6: f is_differentiable_in x by A4, FDIFF_1:9;

cos is_differentiable_in f . x by SIN_COS:63;

hence cos * f is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

for x being Real st x in Z holds

((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b)))

proof

hence
( cos * f is_differentiable_on Z & ( for x being Real st x in Z holds
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 A4, FDIFF_1:9;

then diff ((cos * f),x) = (diff (cos,(f . x))) * (diff (f,x)) by A9, FDIFF_2:13

.= - ((sin . (f . x)) * (diff (f,x))) by A8

.= - ((sin . ((a * x) + b)) * (diff (f,x))) by A2, A10

.= - ((sin . ((a * x) + b)) * ((f `| Z) . x)) by A4, A10, FDIFF_1:def 7

.= - (a * (sin . ((a * x) + b))) by A2, A3, A10, FDIFF_1:23 ;

hence ((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) by A7, A10, FDIFF_1:def 7; :: thesis: verum

end;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 A4, FDIFF_1:9;

then diff ((cos * f),x) = (diff (cos,(f . x))) * (diff (f,x)) by A9, FDIFF_2:13

.= - ((sin . (f . x)) * (diff (f,x))) by A8

.= - ((sin . ((a * x) + b)) * (diff (f,x))) by A2, A10

.= - ((sin . ((a * x) + b)) * ((f `| Z) . x)) by A4, A10, FDIFF_1:def 7

.= - (a * (sin . ((a * x) + b))) by A2, A3, A10, FDIFF_1:23 ;

hence ((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) by A7, A10, FDIFF_1:def 7; :: thesis: verum

((cos * f) `| Z) . x = - (a * (sin . ((a * x) + b))) ) ) by A1, A5, FDIFF_1:9; :: thesis: verum