let Z be open Subset of REAL ; :: thesis: ( not 0 in Z & Z c= dom (sec * ((id Z) ^ )) implies ( sec * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * ((id Z) ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (sec * ((id Z) ^ )) ; :: thesis: ( sec * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * ((id Z) ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) )
A3: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4;
dom (sec * ((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 sec by A2, FUNCT_1:21;
hence cos . (((id Z) ^ ) . x) <> 0 by RFUNCT_1:13; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies sec * ((id Z) ^ ) is_differentiable_in x )
assume A7: x in Z ; :: thesis: sec * ((id Z) ^ ) is_differentiable_in x
then cos . (((id Z) ^ ) . x) <> 0 by A5;
then A8: sec is_differentiable_in ((id Z) ^ ) . x by Th1;
(id Z) ^ is_differentiable_in x by A3, A7, FDIFF_1:16;
hence sec * ((id Z) ^ ) is_differentiable_in x by A8, FDIFF_2:13; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((sec * ((id Z) ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) )
assume A10: x in Z ; :: thesis: ((sec * ((id Z) ^ )) `| Z) . x = - ((sin . (1 / x)) / ((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 sec is_differentiable_in ((id Z) ^ ) . x by Th1;
then diff (sec * ((id Z) ^ )),x = (diff sec ,(((id Z) ^ ) . x)) * (diff ((id Z) ^ ),x) by A11, FDIFF_2:13
.= ((sin . (((id Z) ^ ) . x)) / ((cos . (((id Z) ^ ) . x)) ^2 )) * (diff ((id Z) ^ ),x) by A12, Th1
.= (diff ((id Z) ^ ),x) * ((sin . (((id Z) ^ ) . x)) / ((cos . (((id Z) . x) " )) ^2 )) by A4, A10, RFUNCT_1:def 8
.= (diff ((id Z) ^ ),x) * ((sin . (((id Z) . x) " )) / ((cos . (((id Z) . x) " )) ^2 )) by A4, A10, RFUNCT_1:def 8
.= (diff ((id Z) ^ ),x) * ((sin . (((id Z) . x) " )) / ((cos . (1 * (x " ))) ^2 )) by A10, FUNCT_1:35
.= (diff ((id Z) ^ ),x) * ((sin . (1 * (x " ))) / ((cos . (1 * (x " ))) ^2 )) by A10, FUNCT_1:35
.= ((((id Z) ^ ) `| Z) . x) * ((sin . (1 * (x " ))) / ((cos . (1 * (x " ))) ^2 )) by A3, A10, FDIFF_1:def 8
.= (- (1 / (x ^2 ))) * ((sin . (1 * (x " ))) / ((cos . (1 * (x " ))) ^2 )) by A1, A10, FDIFF_5:4
.= ((- 1) / (x ^2 )) * ((sin . (1 / x)) / ((cos . (1 / x)) ^2 ))
.= ((- 1) * (sin . (1 / x))) / ((x ^2 ) * ((cos . (1 / x)) ^2 )) by XCMPLX_1:77
.= - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ;
hence ((sec * ((id Z) ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) by A9, A10, FDIFF_1:def 8; :: thesis: verum

end;

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