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) ) ) )
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 4;
then A1: Z c= dom (tan - cot ) by XBOOLE_1:18;
then A2: ( tan - cot is_differentiable_on Z & ( for x being Real st x in Z holds
((tan - cot ) `| Z) . x = (1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 )) ) ) by Th5;
A3: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
A4: for x being Real st x in Z holds
exp_R . x <> 0 by SIN_COS:59;
then A5: (tan - cot ) / exp_R is_differentiable_on Z by A2, A3, 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) )
assume A6: 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)
A7: exp_R is_differentiable_in x by SIN_COS:70;
A8: tan - cot is_differentiable_in x by A2, A6, FDIFF_1:16;
A9: (tan - cot ) . x = (tan . x) - (cot . x) by A1, A6, VALUED_1:13;
A10: exp_R . x <> 0 by SIN_COS:59;
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 A7, A8, FDIFF_2:14
.= (((((tan - cot ) `| Z) . x) * (exp_R . x)) - ((diff exp_R ,x) * ((tan - cot ) . x))) / ((exp_R . x) ^2 ) by A2, A6, FDIFF_1:def 8
.= ((((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))) * (exp_R . x)) - ((diff exp_R ,x) * ((tan - cot ) . x))) / ((exp_R . x) ^2 ) by A1, A6, 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:70
.= (((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_1:79
.= (((1 / ((cos . x) ^2 )) + (1 / ((sin . x) ^2 ))) - ((tan . x) - (cot . x))) * (1 / (exp_R . x)) by A10, 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, A6, FDIFF_1:def 8; :: 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 A2, A3, A4, FDIFF_2:21; :: thesis: verum