let Z be open Subset of REAL; :: thesis: ( Z c= dom () implies ( cos * tan is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2)) ) ) )

assume A1: Z c= dom () ; :: thesis: ( cos * tan is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2)) ) )

A2: for x being Real st x in Z holds
cos . x <> 0
proof
let x be Real; :: thesis: ( x in Z implies cos . x <> 0 )
assume x in Z ; :: thesis:
then x in dom () by ;
hence cos . x <> 0 by FDIFF_8:1; :: thesis: verum
end;
A3: for x being Real st x in Z holds
cos * tan is_differentiable_in x
proof end;
then A5: cos * tan is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2)) )
A6: cos is_differentiable_in tan . x by SIN_COS:63;
assume A7: x in Z ; :: thesis: (() `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2))
then A8: cos . x <> 0 by A2;
then tan is_differentiable_in x by FDIFF_7:46;
then diff ((),x) = (diff (cos,(tan . x))) * (diff (tan,x)) by
.= (- (sin (tan . x))) * (diff (tan,x)) by SIN_COS:63
.= (- (sin (tan . x))) * (1 / ((cos . x) ^2)) by
.= - ((sin (tan . x)) / ((cos . x) ^2)) ;
hence ((cos * tan) `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2)) by ; :: thesis: verum
end;
hence ( cos * tan is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = - ((sin (tan . x)) / ((cos . x) ^2)) ) ) by ; :: thesis: verum