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 = x ) holds
( tan * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * (f ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^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 = x ) implies ( tan * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * (f ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) ) )
assume that
A1: Z c= dom (tan * (f ^ )) and
A2: for x being Real st x in Z holds
f . x = x ; :: thesis: ( tan * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * (f ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) )
dom (tan * (f ^ )) c= dom (f ^ ) by RELAT_1:44;
then A3: Z c= dom (f ^ ) by A1, XBOOLE_1:1;
dom (f ^ ) c= dom f by RFUNCT_1:11;
then A4: Z c= dom f by A3, XBOOLE_1:1;
A5: for x being Real st x in Z holds
( f . x = x & f . x <> 0 ) by A2, A3, RFUNCT_1:13;
then A6: ( f ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((f ^ ) `| Z) . x = - (1 / (x ^2 )) ) ) by A4, FDIFF_5:4;
A7: 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;

A8: 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 A9: x in Z ; :: thesis: tan * (f ^ ) is_differentiable_in x
then A10: f ^ is_differentiable_in x by A6, FDIFF_1:16;
cos . ((f ^ ) . x) <> 0 by A7, A9;
then tan is_differentiable_in (f ^ ) . x by FDIFF_7:46;
hence tan * (f ^ ) is_differentiable_in x by A10, FDIFF_2:13; :: thesis: verum

end;

then A11: tan * (f ^ ) is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((tan * (f ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((tan * (f ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) )
assume A12: x in Z ; :: thesis: ((tan * (f ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 )))
then A13: f ^ is_differentiable_in x by A6, FDIFF_1:16;
A14: cos . ((f ^ ) . x) <> 0 by A7, A12;
then tan is_differentiable_in (f ^ ) . x by FDIFF_7:46;
then diff (tan * (f ^ )),x = (diff tan ,((f ^ ) . x)) * (diff (f ^ ),x) by A13, FDIFF_2:13
.= (1 / ((cos . ((f ^ ) . x)) ^2 )) * (diff (f ^ ),x) by A14, FDIFF_7:46
.= (diff (f ^ ),x) / ((cos . ((f . x) " )) ^2 ) by A3, A12, RFUNCT_1:def 8
.= (diff (f ^ ),x) / ((cos . (1 * (x " ))) ^2 ) by A2, A12
.= (((f ^ ) `| Z) . x) / ((cos . (1 * (x " ))) ^2 ) by A6, A12, FDIFF_1:def 8
.= (- (1 / (x ^2 ))) / ((cos . (1 * (x " ))) ^2 ) by A4, A5, A12, FDIFF_5:4
.= ((- 1) / (x ^2 )) / ((cos . (1 / x)) ^2 )
.= (- 1) / ((x ^2 ) * ((cos . (1 / x)) ^2 )) by XCMPLX_1:79
.= - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ;
hence ((tan * (f ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) by A11, A12, 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 = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) ) by A1, A8, FDIFF_1:16; :: thesis: verum