let Z be open Subset of REAL ; :: thesis: ( not 0 in Z & Z c= dom (tan * ((id Z) ^ )) implies ( tan * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * ((id Z) ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (tan * ((id Z) ^ )) ; :: thesis: ( tan * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * ((id Z) ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) )
A3: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4;
dom (tan * ((id Z) ^ )) c= dom ((id Z) ^ ) by RELAT_1:44;
then A4: Z c= dom ((id Z) ^ ) by A2, XBOOLE_1:1;
A5: for x being Real st x in Z holds
cos . (((id Z) ^ ) . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies cos . (((id Z) ^ ) . x) <> 0 )
assume x in Z ; :: thesis: cos . (((id Z) ^ ) . x) <> 0
then ((id Z) ^ ) . x in dom (sin / cos ) by A2, FUNCT_1:21;
hence cos . (((id Z) ^ ) . x) <> 0 by Th1; :: thesis: verum

end;

A6: for x being Real st x in Z holds
tan * ((id Z) ^ ) is_differentiable_in x

proof

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

end;

then A9: tan * ((id Z) ^ ) is_differentiable_on Z by A2, FDIFF_1:16;
for x being Real st x in Z holds
((tan * ((id Z) ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((tan * ((id Z) ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) )
assume A10: x in Z ; :: thesis: ((tan * ((id Z) ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 )))
then A11: (id Z) ^ is_differentiable_in x by A3, FDIFF_1:16;
A12: cos . (((id Z) ^ ) . x) <> 0 by A5, A10;
then tan is_differentiable_in ((id Z) ^ ) . x by FDIFF_7:46;
then diff (tan * ((id Z) ^ )),x = (diff tan ,(((id Z) ^ ) . x)) * (diff ((id Z) ^ ),x) by A11, FDIFF_2:13
.= (1 / ((cos . (((id Z) ^ ) . x)) ^2 )) * (diff ((id Z) ^ ),x) by A12, FDIFF_7:46
.= (diff ((id Z) ^ ),x) / ((cos . (((id Z) . x) " )) ^2 ) by A4, A10, RFUNCT_1:def 8
.= (diff ((id Z) ^ ),x) / ((cos . (1 * (x " ))) ^2 ) by A10, FUNCT_1:35
.= ((((id Z) ^ ) `| Z) . x) / ((cos . (1 * (x " ))) ^2 ) by A3, A10, FDIFF_1:def 8
.= (- (1 / (x ^2 ))) / ((cos . (1 * (x " ))) ^2 ) by A1, A10, 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 * ((id Z) ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) by A9, A10, FDIFF_1:def 8; :: thesis: verum

end;

hence ( tan * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * ((id Z) ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) ) by A2, A6, FDIFF_1:16; :: thesis: verum