let Z be open Subset of REAL ; :: thesis: ( Z c= dom (exp_R * (tan - cot )) implies ( exp_R * (tan - cot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (tan - cot )) `| Z) . x = (exp_R . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))) ) ) )
assume A1: Z c= dom (exp_R * (tan - cot )) ; :: thesis: ( exp_R * (tan - cot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (tan - cot )) `| Z) . x = (exp_R . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))) ) )
dom (exp_R * (tan - cot )) c= dom (tan - cot ) by RELAT_1:44;
then A2: Z c= dom (tan - cot ) by A1, XBOOLE_1:1;
then A3: tan - cot is_differentiable_on Z by Th5;
A4: for x being Real st x in Z holds
exp_R * (tan - cot ) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies exp_R * (tan - cot ) is_differentiable_in x )
assume x in Z ; :: thesis: exp_R * (tan - cot ) is_differentiable_in x
then A5: tan - cot is_differentiable_in x by A3, FDIFF_1:16;
exp_R is_differentiable_in (tan - cot ) . x by SIN_COS:70;
hence exp_R * (tan - cot ) is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

then A6: exp_R * (tan - cot ) is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((exp_R * (tan - cot )) `| Z) . x = (exp_R . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R * (tan - cot )) `| Z) . x = (exp_R . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))) )
assume A7: x in Z ; :: thesis: ((exp_R * (tan - cot )) `| Z) . x = (exp_R . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )))
then A8: tan - cot is_differentiable_in x by A3, FDIFF_1:16;
exp_R is_differentiable_in (tan - cot ) . x by SIN_COS:70;
then diff (exp_R * (tan - cot )),x = (diff exp_R ,((tan - cot ) . x)) * (diff (tan - cot ),x) by A8, FDIFF_2:13
.= (exp_R . ((tan - cot ) . x)) * (diff (tan - cot ),x) by SIN_COS:70
.= (exp_R . ((tan - cot ) . x)) * (((tan - cot ) `| Z) . x) by A3, A7, FDIFF_1:def 8
.= (exp_R . ((tan - cot ) . x)) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))) by A2, A7, Th5
.= (exp_R . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))) by A2, A7, VALUED_1:13 ;
hence ((exp_R * (tan - cot )) `| Z) . x = (exp_R . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))) by A6, A7, FDIFF_1:def 8; :: thesis: verum

end;

hence ( exp_R * (tan - cot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * (tan - cot )) `| Z) . x = (exp_R . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))) ) ) by A1, A4, FDIFF_1:16; :: thesis: verum