let a, b be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom (tan * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( tan * f is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * f) `| Z) . x = a / ((cos . ((a * x) + b)) ^2 ) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom (tan * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( tan * f is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * f) `| Z) . x = a / ((cos . ((a * x) + b)) ^2 ) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (tan * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) implies ( tan * f is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * f) `| Z) . x = a / ((cos . ((a * x) + b)) ^2 ) ) ) )
assume that
A1: Z c= dom (tan * f) and
A2: for x being Real st x in Z holds
f . x = (a * x) + b ; :: thesis: ( tan * f is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * f) `| Z) . x = a / ((cos . ((a * x) + b)) ^2 ) ) )
A3: 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 (sin / cos ) by A1, FUNCT_1:21;
hence cos . (f . x) <> 0 by Th1; :: thesis: verum

end;

dom (tan * f) c= dom f by RELAT_1:44;
then A4: Z c= dom f by A1, XBOOLE_1:1;
then A5: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = a ) ) by A2, FDIFF_1:31;
A6: for x being Real st x in Z holds
tan * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies tan * f is_differentiable_in x )
assume A7: x in Z ; :: thesis: tan * f is_differentiable_in x
then A8: f is_differentiable_in x by A5, FDIFF_1:16;
cos . (f . x) <> 0 by A3, A7;
then tan is_differentiable_in f . x by FDIFF_7:46;
hence tan * f is_differentiable_in x by A8, FDIFF_2:13; :: thesis: verum

end;

then A9: tan * f is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((tan * f) `| Z) . x = a / ((cos . ((a * x) + b)) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies ((tan * f) `| Z) . x = a / ((cos . ((a * x) + b)) ^2 ) )
assume A10: x in Z ; :: thesis: ((tan * f) `| Z) . x = a / ((cos . ((a * x) + b)) ^2 )
then A11: f is_differentiable_in x by A5, FDIFF_1:16;
A12: cos . (f . x) <> 0 by A3, A10;
then tan is_differentiable_in f . x by FDIFF_7:46;
then diff (tan * f),x = (diff tan ,(f . x)) * (diff f,x) by A11, FDIFF_2:13
.= (1 / ((cos . (f . x)) ^2 )) * (diff f,x) by A12, FDIFF_7:46
.= (diff f,x) / ((cos . ((a * x) + b)) ^2 ) by A2, A10
.= ((f `| Z) . x) / ((cos . ((a * x) + b)) ^2 ) by A5, A10, FDIFF_1:def 8
.= a / ((cos . ((a * x) + b)) ^2 ) by A2, A4, A10, FDIFF_1:31 ;
hence ((tan * f) `| Z) . x = a / ((cos . ((a * x) + b)) ^2 ) by A9, A10, FDIFF_1:def 8; :: thesis: verum

end;

hence ( tan * f is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * f) `| Z) . x = a / ((cos . ((a * x) + b)) ^2 ) ) ) by A1, A6, FDIFF_1:16; :: thesis: verum