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_1:25;
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_1:9;
exp_R is_differentiable_in (tan - cot) . x by SIN_COS:65;
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_1:9;
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))) )
A7: exp_R is_differentiable_in (tan - cot) . x by SIN_COS:65;
assume A8: 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 tan - cot is_differentiable_in x by A3, FDIFF_1:9;
then diff ((exp_R * (tan - cot)),x) = (diff (exp_R,((tan - cot) . x))) * (diff ((tan - cot),x)) by A7, FDIFF_2:13
.= (exp_R . ((tan - cot) . x)) * (diff ((tan - cot),x)) by SIN_COS:65
.= (exp_R . ((tan - cot) . x)) * (((tan - cot) `| Z) . x) by A3, A8, FDIFF_1:def 7
.= (exp_R . ((tan - cot) . x)) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) by A2, A8, Th5
.= (exp_R . ((tan . x) - (cot . x))) * ((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) by A2, A8, 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, A8, FDIFF_1:def 7; :: 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_1:9; :: thesis: verum