let Z be open Subset of REAL; ( not 0 in Z & Z c= dom ((id Z) (#) (arctan * ((id Z) ^))) & ( for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 ) ) implies ( (id Z) (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) ) )
set f = id Z;
assume that
A1:
not 0 in Z
and
A2:
Z c= dom ((id Z) (#) (arctan * ((id Z) ^)))
and
A3:
for x being Real st x in Z holds
( ((id Z) ^) . x > - 1 & ((id Z) ^) . x < 1 )
; ( (id Z) (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) )
A4:
Z c= (dom (id Z)) /\ (dom (arctan * ((id Z) ^)))
by A2, VALUED_1:def 4;
then A5:
Z c= dom (arctan * ((id Z) ^))
by XBOOLE_1:18;
then A6:
arctan * ((id Z) ^) is_differentiable_on Z
by A1, A3, SIN_COS9:111;
A7:
for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0
by FUNCT_1:18;
A8:
Z c= dom (id Z)
by A4, XBOOLE_1:18;
then A9:
id Z is_differentiable_on Z
by A7, FDIFF_1:23;
for y being object st y in Z holds
y in dom ((id Z) ^)
by A5, FUNCT_1:11;
then A10:
Z c= dom ((id Z) ^)
by TARSKI:def 3;
for x being Real st x in Z holds
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2)))
proof
let x be
Real;
( x in Z implies (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) )
assume A11:
x in Z
;
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2)))
then (((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x =
(((arctan * ((id Z) ^)) . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x)))
by A2, A6, A9, FDIFF_1:21
.=
(((arctan * ((id Z) ^)) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x)))
by A9, A11, FDIFF_1:def 7
.=
(((arctan * ((id Z) ^)) . x) * 1) + (((id Z) . x) * (diff ((arctan * ((id Z) ^)),x)))
by A8, A7, A11, FDIFF_1:23
.=
(((arctan * ((id Z) ^)) . x) * 1) + (x * (diff ((arctan * ((id Z) ^)),x)))
by A11, FUNCT_1:18
.=
((arctan * ((id Z) ^)) . x) + (x * (((arctan * ((id Z) ^)) `| Z) . x))
by A6, A11, FDIFF_1:def 7
.=
((arctan * ((id Z) ^)) . x) + (x * (- (1 / (1 + (x ^2)))))
by A1, A3, A5, A11, SIN_COS9:111
.=
(arctan . (((id Z) ^) . x)) - (x / (1 + (x ^2)))
by A5, A11, FUNCT_1:12
.=
(arctan . (((id Z) . x) ")) - (x / (1 + (x ^2)))
by A10, A11, RFUNCT_1:def 2
.=
(arctan . (1 / x)) - (x / (1 + (x ^2)))
by A11, FUNCT_1:18
;
hence
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2)))
;
verum
end;
hence
( (id Z) (#) (arctan * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * ((id Z) ^))) `| Z) . x = (arctan . (1 / x)) - (x / (1 + (x ^2))) ) )
by A2, A6, A9, FDIFF_1:21; verum