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_1:25;
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_1:11;
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_1:9;
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_1:9;
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_1:9;
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 2
.= (diff (((id Z) ^),x)) / ((cos . (1 * (x "))) ^2) by A10, FUNCT_1:18
.= ((((id Z) ^) `| Z) . x) / ((cos . (1 * (x "))) ^2) by A3, A10, FDIFF_1:def 7
.= (- (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_1:78
.= - (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 7; :: 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_1:9; :: thesis: verum