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_1:26, SIN_COS:67;
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:63;
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 7
.= - ((diff (cos,x)) / ((cos . x) ^2)) by A6, A4, FDIFF_2:15
.= - ((- (sin . x)) / ((cos . x) ^2)) by SIN_COS:63
.= - (- ((sin . x) / ((cos . x) ^2))) by XCMPLX_1:187
.= (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