let Z be open Subset of REAL ; :: thesis: ( ( for x being Real st x in Z holds
cos . x <> 0 ) implies ( cos ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((cos ^ ) `| Z) . x = (sin . x) / ((cos . x) ^2 ) ) ) )
assume A1: for x being Real st x in Z holds
cos . x <> 0 ; :: thesis: ( cos ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((cos ^ ) `| Z) . x = (sin . x) / ((cos . x) ^2 ) ) )
A2: cos is_differentiable_on Z by FDIFF_1:34, SIN_COS:72;
then A3: cos ^ is_differentiable_on Z by A1, FDIFF_2:22;
for x being Real st x in Z holds
((cos ^ ) `| Z) . x = (sin . x) / ((cos . x) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies ((cos ^ ) `| Z) . x = (sin . x) / ((cos . x) ^2 ) )
A4: cos is_differentiable_in x by SIN_COS:68;
assume A5: x in Z ; :: thesis: ((cos ^ ) `| Z) . x = (sin . x) / ((cos . x) ^2 )
then A6: cos . x <> 0 by A1;
((cos ^ ) `| Z) . x = diff (cos ^ ),x by A3, A5, FDIFF_1:def 8
.= - ((diff cos ,x) / ((cos . x) ^2 )) by A6, A4, FDIFF_2:15
.= - ((- (sin . x)) / ((cos . x) ^2 )) by SIN_COS:68
.= - (- ((sin . x) / ((cos . x) ^2 ))) by XCMPLX_1:188
.= (sin . x) / ((cos . x) ^2 ) ;
hence ((cos ^ ) `| Z) . x = (sin . x) / ((cos . x) ^2 ) ; :: thesis: verum

end;

hence ( cos ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((cos ^ ) `| Z) . x = (sin . x) / ((cos . x) ^2 ) ) ) by A1, A2, FDIFF_2:22; :: thesis: verum