let Z be open Subset of REAL; :: thesis: ( 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: 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)) ) )
dom (tan * ((id Z) ^)) c= dom ((id Z) ^) by RELAT_1:44;
then A2: Z c= dom ((id Z) ^) by XBOOLE_1:1, A1;
A3: not 0 in Z

proof

assume K: 0 in Z ; :: thesis: contradiction
dom ((id Z) ^) = (dom (id Z)) \ ((id Z) " {0}) by RFUNCT_1:def 8
.= (dom (id Z)) \ {0} by Lm0, K ;
then not 0 in {0} by XBOOLE_0:def 5, K, A2;
hence contradiction by TARSKI:def 1; :: thesis: verum

end;

A4: Z c= dom (- (tan * ((id Z) ^))) by A1, VALUED_1:8;
B1: tan * ((id Z) ^) is_differentiable_on Z by A1, A3, FDIFF_8:8;
then B2: (- 1) (#) (tan * ((id Z) ^)) is_differentiable_on Z by A4, FDIFF_1:28, A;
A5: ( (id Z) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) ^) `| Z) . x = - (1 / (x ^2)) ) ) by A3, FDIFF_5:4;
A6: 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 FDIFF_8:1; :: thesis: verum

end;

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 A7: x in Z ; :: thesis: ((- (tan * ((id Z) ^))) `| Z) . x = 1 / ((x ^2) * ((cos . (1 / x)) ^2))
then A8: (id Z) ^ is_differentiable_in x by A5, FDIFF_1:16;
A9: cos . (((id Z) ^) . x) <> 0 by A6, A7;
then A10: tan is_differentiable_in ((id Z) ^) . x by FDIFF_7:46;
A11: tan * ((id Z) ^) is_differentiable_in x by B1, A7, FDIFF_1:16;
((- (tan * ((id Z) ^))) `| Z) . x = diff ((- (tan * ((id Z) ^))),x) by B2, A7, FDIFF_1:def 8
.= (- 1) * (diff ((tan * ((id Z) ^)),x)) by A11, FDIFF_1:23, A
.= (- 1) * ((diff (tan,(((id Z) ^) . x))) * (diff (((id Z) ^),x))) by A8, A10, FDIFF_2:13
.= (- 1) * ((1 / ((cos . (((id Z) ^) . x)) ^2)) * (diff (((id Z) ^),x))) by A9, FDIFF_7:46
.= (- 1) * ((diff (((id Z) ^),x)) / ((cos . (((id Z) . x) ")) ^2)) by A2, A7, RFUNCT_1:def 8
.= (- 1) * ((diff (((id Z) ^),x)) / ((cos . (1 * (x "))) ^2)) by FUNCT_1:35, A7
.= (- 1) * (((((id Z) ^) `| Z) . x) / ((cos . (1 * (x "))) ^2)) by A5, A7, FDIFF_1:def 8
.= (- 1) * ((- (1 / (x ^2))) / ((cos . (1 * (x "))) ^2)) by A7, A3, FDIFF_5:4
.= (- 1) * (((- 1) / (x ^2)) / ((cos . (1 / x)) ^2))
.= (- 1) * ((- 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)) ; :: 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 B2; :: thesis: verum