let a, b be Real; :: thesis: for Z being open Subset of REAL

for f being PartFunc of REAL,REAL st Z c= dom (sec * f) & ( for x being Real st x in Z holds

f . x = (a * x) + b ) holds

( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom (sec * f) & ( for x being Real st x in Z holds

f . x = (a * x) + b ) holds

( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (sec * f) & ( for x being Real st x in Z holds

f . x = (a * x) + b ) implies ( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) ) )

assume that

A1: Z c= dom (sec * f) and

A2: for x being Real st x in Z holds

f . x = (a * x) + b ; :: thesis: ( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) )

dom (sec * f) c= dom f by RELAT_1:25;

then A3: Z c= dom f by A1, XBOOLE_1:1;

then A4: f is_differentiable_on Z by A2, FDIFF_1:23;

A5: for x being Real st x in Z holds

cos . (f . x) <> 0

sec * f is_differentiable_in x

for x being Real st x in Z holds

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2)

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum

for f being PartFunc of REAL,REAL st Z c= dom (sec * f) & ( for x being Real st x in Z holds

f . x = (a * x) + b ) holds

( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom (sec * f) & ( for x being Real st x in Z holds

f . x = (a * x) + b ) holds

( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (sec * f) & ( for x being Real st x in Z holds

f . x = (a * x) + b ) implies ( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) ) )

assume that

A1: Z c= dom (sec * f) and

A2: for x being Real st x in Z holds

f . x = (a * x) + b ; :: thesis: ( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) )

dom (sec * f) c= dom f by RELAT_1:25;

then A3: Z c= dom f by A1, XBOOLE_1:1;

then A4: f is_differentiable_on Z by A2, FDIFF_1:23;

A5: for x being Real st x in Z holds

cos . (f . x) <> 0

proof

A6:
for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies cos . (f . x) <> 0 )

assume x in Z ; :: thesis: cos . (f . x) <> 0

then f . x in dom sec by A1, FUNCT_1:11;

hence cos . (f . x) <> 0 by RFUNCT_1:3; :: thesis: verum

end;assume x in Z ; :: thesis: cos . (f . x) <> 0

then f . x in dom sec by A1, FUNCT_1:11;

hence cos . (f . x) <> 0 by RFUNCT_1:3; :: thesis: verum

sec * f is_differentiable_in x

proof

then A9:
sec * f is_differentiable_on Z
by A1, FDIFF_1:9;
let x be Real; :: thesis: ( x in Z implies sec * f is_differentiable_in x )

assume A7: x in Z ; :: thesis: sec * f is_differentiable_in x

then cos . (f . x) <> 0 by A5;

then A8: sec is_differentiable_in f . x by Th1;

f is_differentiable_in x by A4, A7, FDIFF_1:9;

hence sec * f is_differentiable_in x by A8, FDIFF_2:13; :: thesis: verum

end;assume A7: x in Z ; :: thesis: sec * f is_differentiable_in x

then cos . (f . x) <> 0 by A5;

then A8: sec is_differentiable_in f . x by Th1;

f is_differentiable_in x by A4, A7, FDIFF_1:9;

hence sec * f is_differentiable_in x by A8, FDIFF_2:13; :: thesis: verum

for x being Real st x in Z holds

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2)

proof

hence
( sec * f is_differentiable_on Z & ( for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies ((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) )

assume A10: x in Z ; :: thesis: ((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2)

then A11: f is_differentiable_in x by A4, FDIFF_1:9;

A12: cos . (f . x) <> 0 by A5, A10;

then sec is_differentiable_in f . x by Th1;

then diff ((sec * f),x) = (diff (sec,(f . x))) * (diff (f,x)) by A11, FDIFF_2:13

.= ((sin . (f . x)) / ((cos . (f . x)) ^2)) * (diff (f,x)) by A12, Th1

.= (diff (f,x)) * ((sin . (f . x)) / ((cos . ((a * x) + b)) ^2)) by A2, A10

.= ((f `| Z) . x) * ((sin . (f . x)) / ((cos . ((a * x) + b)) ^2)) by A4, A10, FDIFF_1:def 7

.= a * ((sin . (f . x)) / ((cos . ((a * x) + b)) ^2)) by A2, A3, A10, FDIFF_1:23

.= a * ((sin . ((a * x) + b)) / ((cos . ((a * x) + b)) ^2)) by A2, A10 ;

hence ((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) by A9, A10, FDIFF_1:def 7; :: thesis: verum

end;assume A10: x in Z ; :: thesis: ((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2)

then A11: f is_differentiable_in x by A4, FDIFF_1:9;

A12: cos . (f . x) <> 0 by A5, A10;

then sec is_differentiable_in f . x by Th1;

then diff ((sec * f),x) = (diff (sec,(f . x))) * (diff (f,x)) by A11, FDIFF_2:13

.= ((sin . (f . x)) / ((cos . (f . x)) ^2)) * (diff (f,x)) by A12, Th1

.= (diff (f,x)) * ((sin . (f . x)) / ((cos . ((a * x) + b)) ^2)) by A2, A10

.= ((f `| Z) . x) * ((sin . (f . x)) / ((cos . ((a * x) + b)) ^2)) by A4, A10, FDIFF_1:def 7

.= a * ((sin . (f . x)) / ((cos . ((a * x) + b)) ^2)) by A2, A3, A10, FDIFF_1:23

.= a * ((sin . ((a * x) + b)) / ((cos . ((a * x) + b)) ^2)) by A2, A10 ;

hence ((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) by A9, A10, FDIFF_1:def 7; :: thesis: verum

((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum