let Z be open Subset of REAL; :: thesis: ( Z c= ].(- 1),1.[ implies ( (arctan - arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) ) )
assume A1: Z c= ].(- 1),1.[ ; :: thesis: ( (arctan - arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) )
then A2: arctan - arccot is_differentiable_on Z by Th38;
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A1, XBOOLE_1:1;
].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then Z c= dom arctan by A1, XBOOLE_1:1;
then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A5: Z c= dom (arctan - arccot) by VALUED_1:12;
A6: ( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
exp_R . x <> 0 ) ) by FDIFF_1:26, SIN_COS:54, TAYLOR_1:16;
then A7: (arctan - arccot) / exp_R is_differentiable_on Z by A2, FDIFF_2:21;
for x being Real st x in Z holds
(((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x)

proof

let x be Real; :: thesis: ( x in Z implies (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) )
A8: exp_R is_differentiable_in x by SIN_COS:65;
A9: exp_R . x <> 0 by SIN_COS:54;
assume A10: x in Z ; :: thesis: (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x)
then A11: arctan - arccot is_differentiable_in x by A2, FDIFF_1:9;
(((arctan - arccot) / exp_R) `| Z) . x = diff (((arctan - arccot) / exp_R),x) by A7, A10, FDIFF_1:def 7
.= (((diff ((arctan - arccot),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by A11, A8, A9, FDIFF_2:14
.= (((((arctan - arccot) `| Z) . x) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by A2, A10, FDIFF_1:def 7
.= (((2 / (1 + (x ^2))) * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by A1, A10, Th38
.= (((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan - arccot) . x))) / ((exp_R . x) ^2) by SIN_COS:65
.= (((2 / (1 + (x ^2))) * (exp_R . x)) - ((exp_R . x) * ((arctan . x) - (arccot . x)))) / ((exp_R . x) ^2) by A5, A10, VALUED_1:13
.= ((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * ((exp_R . x) / ((exp_R . x) * (exp_R . x)))
.= ((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)) by XCMPLX_1:78
.= ((2 / (1 + (x ^2))) - ((arctan . x) - (arccot . x))) * (1 / (exp_R . x)) by A9, XCMPLX_1:60
.= (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ;
hence (((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ; :: thesis: verum

end;

hence ( (arctan - arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan - arccot) / exp_R) `| Z) . x = (((2 / (1 + (x ^2))) - (arctan . x)) + (arccot . x)) / (exp_R . x) ) ) by A2, A6, FDIFF_2:21; :: thesis: verum