let Z be open Subset of REAL; :: thesis: ( Z c= dom ((tan - cot) / exp_R) implies ( (tan - cot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((tan - cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) - (tan . x)) + (cot . x)) / (exp_R . x) ) ) )
A1: for x being Real st x in Z holds
exp_R . x <> 0 by SIN_COS:54;
assume Z c= dom ((tan - cot) / exp_R) ; :: thesis: ( (tan - cot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((tan - cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) - (tan . x)) + (cot . x)) / (exp_R . x) ) )
then Z c= (dom (tan - cot)) /\ ((dom exp_R) \ (exp_R " {0})) by RFUNCT_1:def 1;
then A2: Z c= dom (tan - cot) by XBOOLE_1:18;
then A3: tan - cot is_differentiable_on Z by Th5;
A4: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
then A5: (tan - cot) / exp_R is_differentiable_on Z by A3, A1, FDIFF_2:21;
for x being Real st x in Z holds
(((tan - cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) - (tan . x)) + (cot . x)) / (exp_R . x)

proof

let x be Real; :: thesis: ( x in Z implies (((tan - cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) - (tan . x)) + (cot . x)) / (exp_R . x) )
A6: exp_R is_differentiable_in x by SIN_COS:65;
A7: exp_R . x <> 0 by SIN_COS:54;
assume A8: x in Z ; :: thesis: (((tan - cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) - (tan . x)) + (cot . x)) / (exp_R . x)
then A9: (tan - cot) . x = (tan . x) - (cot . x) by A2, VALUED_1:13;
tan - cot is_differentiable_in x by A3, A8, FDIFF_1:9;
then diff (((tan - cot) / exp_R),x) = (((diff ((tan - cot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((tan - cot) . x))) / ((exp_R . x) ^2) by A6, A7, FDIFF_2:14
.= (((((tan - cot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((tan - cot) . x))) / ((exp_R . x) ^2) by A3, A8, FDIFF_1:def 7
.= ((((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) * (exp_R . x)) - ((diff (exp_R,x)) * ((tan - cot) . x))) / ((exp_R . x) ^2) by A2, A8, Th5
.= ((((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) * (exp_R . x)) - ((exp_R . x) * ((tan . x) - (cot . x)))) / ((exp_R . x) * (exp_R . x)) by A9, SIN_COS:65
.= (((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) - ((tan . x) - (cot . x))) * ((exp_R . x) / ((exp_R . x) * (exp_R . x)))
.= (((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) - ((tan . x) - (cot . x))) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)) by XCMPLX_1:78
.= (((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) - ((tan . x) - (cot . x))) * (1 / (exp_R . x)) by A7, XCMPLX_1:60
.= (((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) - ((tan . x) - (cot . x))) / (exp_R . x) ;
hence (((tan - cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) - (tan . x)) + (cot . x)) / (exp_R . x) by A5, A8, FDIFF_1:def 7; :: thesis: verum

end;

hence ( (tan - cot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((tan - cot) / exp_R) `| Z) . x = ((((1 / ((cos . x) ^2)) + (1 / ((sin . x) ^2))) - (tan . x)) + (cot . x)) / (exp_R . x) ) ) by A3, A4, A1, FDIFF_2:21; :: thesis: verum