let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL , REAL st Z c= dom (sec * (f ^ )) & ( for x being Real st x in Z holds
f . x = x ) holds
( sec * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * (f ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) )
let f be PartFunc of REAL , REAL ; :: thesis: ( Z c= dom (sec * (f ^ )) & ( for x being Real st x in Z holds
f . x = x ) implies ( sec * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * (f ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) ) )
assume that
A1: Z c= dom (sec * (f ^ )) and
A2: for x being Real st x in Z holds
f . x = x ; :: thesis: ( sec * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * (f ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) )
dom (sec * (f ^ )) c= dom (f ^ ) by RELAT_1:44;
then A3: Z c= dom (f ^ ) by A1, XBOOLE_1:1;
dom (f ^ ) c= dom f by RFUNCT_1:11;
then A4: Z c= dom f by A3, XBOOLE_1:1;
A5: for x being Real st x in Z holds
( f . x = x & f . x <> 0 ) by A2, A3, RFUNCT_1:13;
then A6: ( f ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((f ^ ) `| Z) . x = - (1 / (x ^2 )) ) ) by A4, FDIFF_5:4;
A7: for x being Real st x in Z holds
cos . ((f ^ ) . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies cos . ((f ^ ) . x) <> 0 )
assume x in Z ; :: thesis: cos . ((f ^ ) . x) <> 0
then (f ^ ) . x in dom sec by A1, FUNCT_1:21;
hence cos . ((f ^ ) . x) <> 0 by RFUNCT_1:13; :: thesis: verum

end;

A8: for x being Real st x in Z holds
sec * (f ^ ) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies sec * (f ^ ) is_differentiable_in x )
assume A9: x in Z ; :: thesis: sec * (f ^ ) is_differentiable_in x
then A10: f ^ is_differentiable_in x by A6, FDIFF_1:16;
cos . ((f ^ ) . x) <> 0 by A7, A9;
then sec is_differentiable_in (f ^ ) . x by Th1;
hence sec * (f ^ ) is_differentiable_in x by A10, FDIFF_2:13; :: thesis: verum

end;

then A11: sec * (f ^ ) is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((sec * (f ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((sec * (f ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) )
assume A12: x in Z ; :: thesis: ((sec * (f ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 )))
then A13: f ^ is_differentiable_in x by A6, FDIFF_1:16;
A14: cos . ((f ^ ) . x) <> 0 by A7, A12;
then sec is_differentiable_in (f ^ ) . x by Th1;
then diff (sec * (f ^ )),x = (diff sec ,((f ^ ) . x)) * (diff (f ^ ),x) by A13, FDIFF_2:13
.= ((sin . ((f ^ ) . x)) / ((cos . ((f ^ ) . x)) ^2 )) * (diff (f ^ ),x) by A14, Th1
.= (diff (f ^ ),x) * ((sin . ((f ^ ) . x)) / ((cos . ((f . x) " )) ^2 )) by A3, A12, RFUNCT_1:def 8
.= (diff (f ^ ),x) * ((sin . ((f . x) " )) / ((cos . ((f . x) " )) ^2 )) by A3, A12, RFUNCT_1:def 8
.= (diff (f ^ ),x) * ((sin . ((f . x) " )) / ((cos . (1 * (x " ))) ^2 )) by A2, A12
.= (diff (f ^ ),x) * ((sin . (1 * (x " ))) / ((cos . (1 * (x " ))) ^2 )) by A2, A12
.= (((f ^ ) `| Z) . x) * ((sin . (1 * (x " ))) / ((cos . (1 * (x " ))) ^2 )) by A6, A12, FDIFF_1:def 8
.= (- (1 / (x ^2 ))) * ((sin . (1 * (x " ))) / ((cos . (1 * (x " ))) ^2 )) by A4, A5, A12, 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 * (f ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) by A11, A12, FDIFF_1:def 8; :: thesis: verum

end;

hence ( sec * (f ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * (f ^ )) `| Z) . x = - ((sin . (1 / x)) / ((x ^2 ) * ((cos . (1 / x)) ^2 ))) ) ) by A1, A8, FDIFF_1:16; :: thesis: verum