let Z be open Subset of REAL ; :: thesis: ( Z c= dom (sec * exp_R ) implies ( sec * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * exp_R ) `| Z) . x = ((exp_R . x) * (sin . (exp_R . x))) / ((cos . (exp_R . x)) ^2 ) ) ) )
assume A1: Z c= dom (sec * exp_R ) ; :: thesis: ( sec * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * exp_R ) `| Z) . x = ((exp_R . x) * (sin . (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 sec by A1, FUNCT_1:21;
hence cos . (exp_R . x) <> 0 by RFUNCT_1:13; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies sec * exp_R is_differentiable_in x )
assume A4: x in Z ; :: thesis: sec * exp_R is_differentiable_in x
A5: exp_R is_differentiable_in x by SIN_COS:70;
cos . (exp_R . x) <> 0 by A2, A4;
then sec is_differentiable_in exp_R . x by Th1;
hence sec * exp_R is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

then A6: sec * exp_R is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((sec * exp_R ) `| Z) . x = ((exp_R . x) * (sin . (exp_R . x))) / ((cos . (exp_R . x)) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies ((sec * exp_R ) `| Z) . x = ((exp_R . x) * (sin . (exp_R . x))) / ((cos . (exp_R . x)) ^2 ) )
assume A7: x in Z ; :: thesis: ((sec * exp_R ) `| Z) . x = ((exp_R . x) * (sin . (exp_R . x))) / ((cos . (exp_R . x)) ^2 )
A8: exp_R is_differentiable_in x by SIN_COS:70;
A9: cos . (exp_R . x) <> 0 by A2, A7;
then sec is_differentiable_in exp_R . x by Th1;
then diff (sec * exp_R ),x = (diff sec ,(exp_R . x)) * (diff exp_R ,x) by A8, FDIFF_2:13
.= ((sin . (exp_R . x)) / ((cos . (exp_R . x)) ^2 )) * (diff exp_R ,x) by A9, Th1
.= (exp_R . x) * ((sin . (exp_R . x)) / ((cos . (exp_R . x)) ^2 )) by SIN_COS:70 ;
hence ((sec * exp_R ) `| Z) . x = ((exp_R . x) * (sin . (exp_R . x))) / ((cos . (exp_R . x)) ^2 ) by A6, A7, FDIFF_1:def 8; :: thesis: verum

end;

hence ( sec * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((sec * exp_R ) `| Z) . x = ((exp_R . x) * (sin . (exp_R . x))) / ((cos . (exp_R . x)) ^2 ) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum