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 A1: ( not 0 in Z & 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 ))) ) )
A2: for x being Real st x in Z holds
(id Z) . x = x by FUNCT_1:35;
dom (tan * ((id Z) ^ )) c= dom ((id Z) ^ ) by RELAT_1:44;
then A3: Z c= dom ((id Z) ^ ) by A1, XBOOLE_1:1;
A6: ( (id Z) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) ^ ) `| Z) . x = - (1 / (x ^2 )) ) ) by FDIFF_5:4, A1;
A7: 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 A1, FUNCT_1:21;
hence cos . (((id Z) ^ ) . x) <> 0 by Th1; :: thesis: verum

end;

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

end;

then A11: tan * ((id Z) ^ ) is_differentiable_on Z by A1, 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 A12: x in Z ; :: thesis: ((tan * ((id Z) ^ )) `| Z) . x = - (1 / ((x ^2 ) * ((cos . (1 / x)) ^2 )))
then A13: (id Z) ^ is_differentiable_in x by A6, FDIFF_1:16;
A14: cos . (((id Z) ^ ) . x) <> 0 by A7, A12;
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 A13, FDIFF_2:13
.= (1 / ((cos . (((id Z) ^ ) . x)) ^2 )) * (diff ((id Z) ^ ),x) by A14, FDIFF_7:46
.= (diff ((id Z) ^ ),x) / ((cos . (((id Z) . x) " )) ^2 ) by A3, A12, RFUNCT_1:def 8
.= (diff ((id Z) ^ ),x) / ((cos . (1 * (x " ))) ^2 ) by A2, A12
.= ((((id Z) ^ ) `| Z) . x) / ((cos . (1 * (x " ))) ^2 ) by A6, A12, FDIFF_1:def 8
.= (- (1 / (x ^2 ))) / ((cos . (1 * (x " ))) ^2 ) by A12, FDIFF_5:4, A1
.= ((- 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 A11, A12, 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 A1, A8, FDIFF_1:16; :: thesis: verum