let Z be open Subset of REAL; :: thesis: ( Z c= dom () implies ( tan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (1 / ((cos . (tan . x)) ^2)) * (1 / ((cos . x) ^2)) ) ) )

assume A1: Z c= dom () ; :: thesis: ( tan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (1 / ((cos . (tan . x)) ^2)) * (1 / ((cos . x) ^2)) ) )

A2: for x being Real st x in Z holds
cos . (tan . x) <> 0
proof
let x be Real; :: thesis: ( x in Z implies cos . (tan . x) <> 0 )
assume x in Z ; :: thesis: cos . (tan . x) <> 0
then tan . x in dom tan by ;
hence cos . (tan . x) <> 0 by FDIFF_8:1; :: thesis: verum
end;
A3: 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;
A4: for x being Real st x in Z holds
tan * tan is_differentiable_in x
proof end;
then A7: tan * tan is_differentiable_on Z by ;
for x being Real st x in Z holds
(() `| Z) . x = (1 / ((cos . (tan . x)) ^2)) * (1 / ((cos . x) ^2))
proof
let x be Real; :: thesis: ( x in Z implies (() `| Z) . x = (1 / ((cos . (tan . x)) ^2)) * (1 / ((cos . x) ^2)) )
assume A8: x in Z ; :: thesis: (() `| Z) . x = (1 / ((cos . (tan . x)) ^2)) * (1 / ((cos . x) ^2))
then A9: cos . (tan . x) <> 0 by A2;
then A10: tan is_differentiable_in tan . x by FDIFF_7:46;
A11: cos . x <> 0 by A3, A8;
then tan is_differentiable_in x by FDIFF_7:46;
then diff ((),x) = (diff (tan,(tan . x))) * (diff (tan,x)) by
.= (1 / ((cos . (tan . x)) ^2)) * (diff (tan,x)) by
.= (1 / ((cos . (tan . x)) ^2)) * (1 / ((cos . x) ^2)) by ;
hence ((tan * tan) `| Z) . x = (1 / ((cos . (tan . x)) ^2)) * (1 / ((cos . x) ^2)) by ; :: thesis: verum
end;
hence ( tan * tan is_differentiable_on Z & ( for x being Real st x in Z holds
(() `| Z) . x = (1 / ((cos . (tan . x)) ^2)) * (1 / ((cos . x) ^2)) ) ) by ; :: thesis: verum