let Z be open Subset of REAL; :: thesis: ( Z c= ].(- 1),1.[ implies ( exp_R (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) ) )
assume A1: Z c= ].(- 1),1.[ ; :: thesis: ( exp_R (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) )
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;
then Z c= (dom exp_R) /\ (dom (arctan - arccot)) by TAYLOR_1:16, XBOOLE_1:19;
then A6: Z c= dom (exp_R (#) (arctan - arccot)) by VALUED_1:def 4;
A7: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
for x being Real st x in Z holds
((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2)))

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) )
assume A8: x in Z ; :: thesis: ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2)))
then ((exp_R (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan - arccot),x))) by A6, A7, A2, FDIFF_1:21
.= (((arctan . x) - (arccot . x)) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((arctan - arccot),x))) by A5, A8, VALUED_1:13
.= (((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (diff ((arctan - arccot),x))) by TAYLOR_1:16
.= (((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (((arctan - arccot) `| Z) . x)) by A2, A8, FDIFF_1:def 7
.= (((arctan . x) - (arccot . x)) * (exp_R . x)) + ((exp_R . x) * (2 / (1 + (x ^2)))) by A1, A8, Th38
.= ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ;
hence ((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ; :: thesis: verum

end;

hence ( exp_R (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (arctan - arccot)) `| Z) . x = ((exp_R . x) * ((arctan . x) - (arccot . x))) + ((2 * (exp_R . x)) / (1 + (x ^2))) ) ) by A6, A7, A2, FDIFF_1:21; :: thesis: verum