let Z be open Subset of REAL; :: thesis: ( Z c= dom (tan * exp_R) implies ( tan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * exp_R) `| Z) . x = (exp_R . x) / ((cos . (exp_R . x)) ^2) ) ) )
assume A1: Z c= dom (tan * exp_R) ; :: thesis: ( tan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * exp_R) `| Z) . x = (exp_R . x) / ((cos . (exp_R . x)) ^2) ) )
A2: for x being Real st x in Z holds
cos . (exp_R . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies cos . (exp_R . x) <> 0 )
assume x in Z ; :: thesis: cos . (exp_R . x) <> 0
then exp_R . x in dom (sin / cos) by A1, FUNCT_1:11;
hence cos . (exp_R . x) <> 0 by Th1; :: thesis: verum

end;

A3: for x being Real st x in Z holds
tan * exp_R is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies tan * exp_R is_differentiable_in x )
assume x in Z ; :: thesis: tan * exp_R is_differentiable_in x
then cos . (exp_R . x) <> 0 by A2;
then ( exp_R is_differentiable_in x & tan is_differentiable_in exp_R . x ) by FDIFF_7:46, SIN_COS:65;
hence tan * exp_R is_differentiable_in x by FDIFF_2:13; :: thesis: verum

end;

then A4: tan * exp_R is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((tan * exp_R) `| Z) . x = (exp_R . x) / ((cos . (exp_R . x)) ^2)

proof

let x be Real; :: thesis: ( x in Z implies ((tan * exp_R) `| Z) . x = (exp_R . x) / ((cos . (exp_R . x)) ^2) )
assume A5: x in Z ; :: thesis: ((tan * exp_R) `| Z) . x = (exp_R . x) / ((cos . (exp_R . x)) ^2)
then A6: cos . (exp_R . x) <> 0 by A2;
then ( exp_R is_differentiable_in x & tan is_differentiable_in exp_R . x ) by FDIFF_7:46, SIN_COS:65;
then diff ((tan * exp_R),x) = (diff (tan,(exp_R . x))) * (diff (exp_R,x)) by FDIFF_2:13
.= (1 / ((cos . (exp_R . x)) ^2)) * (diff (exp_R,x)) by A6, FDIFF_7:46
.= (exp_R . x) / ((cos . (exp_R . x)) ^2) by SIN_COS:65 ;
hence ((tan * exp_R) `| Z) . x = (exp_R . x) / ((cos . (exp_R . x)) ^2) by A4, A5, FDIFF_1:def 7; :: thesis: verum

end;

hence ( tan * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((tan * exp_R) `| Z) . x = (exp_R . x) / ((cos . (exp_R . x)) ^2) ) ) by A1, A3, FDIFF_1:9; :: thesis: verum