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 . x) ^2) / (cos . x))) + (1 / (cos . x)) ) ) )

A1: for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:63;
assume A2: Z c= dom () ; :: thesis: ( cos (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (- (((sin . x) ^2) / (cos . x))) + (1 / (cos . x)) ) )

then A3: Z c= () /\ () by VALUED_1:def 4;
then A4: Z c= dom tan by XBOOLE_1:18;
for x being Real st x in Z holds
tan is_differentiable_in x
proof end;
then A5: tan is_differentiable_on Z by ;
Z c= dom cos by ;
then A6: cos is_differentiable_on Z by ;
A7: for x being Real st x in Z holds
diff (tan,x) = 1 / ((cos . x) ^2)
proof
let x be Real; :: thesis: ( x in Z implies diff (tan,x) = 1 / ((cos . x) ^2) )
assume x in Z ; :: thesis: diff (tan,x) = 1 / ((cos . x) ^2)
then cos . x <> 0 by ;
hence diff (tan,x) = 1 / ((cos . x) ^2) by FDIFF_7:46; :: thesis: verum
end;
for x being Real st x in Z holds
(() `| Z) . x = (- (((sin . x) ^2) / (cos . x))) + (1 / (cos . x))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = (- (((sin . x) ^2) / (cos . x))) + (1 / (cos . x)) )
assume A8: x in Z ; :: thesis: (() `| Z) . x = (- (((sin . x) ^2) / (cos . x))) + (1 / (cos . x))
then (() `| Z) . x = ((diff (cos,x)) * (tan . x)) + ((cos . x) * (diff (tan,x))) by
.= ((tan . x) * (- (sin . x))) + ((cos . x) * (diff (tan,x))) by SIN_COS:63
.= ((tan . x) * (- (sin . x))) + ((cos . x) * (1 / ((cos . x) ^2))) by A7, A8
.= (((sin . x) / (cos . x)) * (- ((sin . x) / 1))) + ((cos . x) / ((cos . x) ^2)) by
.= (- (((sin . x) ^2) / (cos . x))) + (((cos . x) / (cos . x)) / (cos . x)) by XCMPLX_1:78
.= (- (((sin . x) ^2) / (cos . x))) + (1 / (cos . x)) by ;
hence ((cos (#) tan) `| Z) . x = (- (((sin . x) ^2) / (cos . x))) + (1 / (cos . x)) ; :: thesis: verum
end;
hence ( cos (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (- (((sin . x) ^2) / (cos . x))) + (1 / (cos . x)) ) ) by ; :: thesis: verum