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

proof

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;

A6: for x being Real st x in Z holds
sec * f is_differentiable_in x

proof

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;

then A9: sec * f is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((sec * f) `| Z) . x = (a * (sin . ((a * x) + b))) / ((cos . ((a * x) + b)) ^2)

proof

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;

hence ( 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) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum