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

end;

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

proof

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

end;

then A5: exp_R * sec is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((exp_R * sec ) `| Z) . x = ((exp_R . (sec . x)) * (sin . x)) / ((cos . x) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R * sec ) `| Z) . x = ((exp_R . (sec . x)) * (sin . x)) / ((cos . x) ^2 ) )
A6: exp_R is_differentiable_in sec . x by SIN_COS:70;
assume A7: x in Z ; :: thesis: ((exp_R * sec ) `| Z) . x = ((exp_R . (sec . x)) * (sin . x)) / ((cos . x) ^2 )
then A8: cos . x <> 0 by A2;
then sec is_differentiable_in x by Th1;
then diff (exp_R * sec ),x = (diff exp_R ,(sec . x)) * (diff sec ,x) by A6, FDIFF_2:13
.= (diff exp_R ,(sec . x)) * ((sin . x) / ((cos . x) ^2 )) by A8, Th1
.= (exp_R . (sec . x)) * ((sin . x) / ((cos . x) ^2 )) by SIN_COS:70 ;
hence ((exp_R * sec ) `| Z) . x = ((exp_R . (sec . x)) * (sin . x)) / ((cos . x) ^2 ) by A5, A7, FDIFF_1:def 8; :: thesis: verum

end;

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