let Z be open Subset of REAL; ( 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 = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) ) )
assume A1:
Z c= ].(- 1),1.[
; ( (arctan + arccot) / exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) )
then A2:
arctan + arccot is_differentiable_on Z
by Th37;
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:def 1;
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 = - (((arctan . x) + (arccot . x)) / (exp_R . x))
proof
let x be
Real;
( x in Z implies (((arctan + arccot) / exp_R) `| Z) . x = - (((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
;
(((arctan + arccot) / exp_R) `| Z) . x = - (((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
.=
((0 * (exp_R . x)) - ((diff (exp_R,x)) * ((arctan + arccot) . x))) / ((exp_R . x) ^2)
by A1, A10, Th37
.=
- (((diff (exp_R,x)) * ((arctan + arccot) . x)) / ((exp_R . x) ^2))
.=
- (((exp_R . x) * ((arctan + arccot) . x)) / ((exp_R . x) ^2))
by SIN_COS:65
.=
- ((((arctan . x) + (arccot . x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x)))
by A5, A10, VALUED_1:def 1
.=
- (((arctan . x) + (arccot . x)) * ((exp_R . x) / ((exp_R . x) * (exp_R . x))))
.=
- (((arctan . x) + (arccot . x)) * (((exp_R . x) / (exp_R . x)) / (exp_R . x)))
by XCMPLX_1:78
.=
- (((arctan . x) + (arccot . x)) * (1 / (exp_R . x)))
by A9, XCMPLX_1:60
.=
- (((arctan . x) + (arccot . x)) / (exp_R . x))
;
hence
(((arctan + arccot) / exp_R) `| Z) . x = - (((arctan . x) + (arccot . x)) / (exp_R . x))
;
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 = - (((arctan . x) + (arccot . x)) / (exp_R . x)) ) )
by A2, A6, FDIFF_2:21; verum