let Z be open Subset of REAL; ( Z c= dom ((#Z 2) * arccot) & Z c= ].(- 1),1.[ implies ( - ((1 / 2) (#) ((#Z 2) * arccot)) is_differentiable_on Z & ( for x being Real st x in Z holds
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = (arccot . x) / (1 + (x ^2)) ) ) )
assume A1:
( Z c= dom ((#Z 2) * arccot) & Z c= ].(- 1),1.[ )
; ( - ((1 / 2) (#) ((#Z 2) * arccot)) is_differentiable_on Z & ( for x being Real st x in Z holds
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = (arccot . x) / (1 + (x ^2)) ) )
then A2:
Z c= dom ((1 / 2) (#) ((#Z 2) * arccot))
by VALUED_1:def 5;
then A3:
Z c= dom (- ((1 / 2) (#) ((#Z 2) * arccot)))
by VALUED_1:8;
A4:
(1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z
by A2, A1, SIN_COS9:94;
then A5:
(- 1) (#) ((1 / 2) (#) ((#Z 2) * arccot)) is_differentiable_on Z
by A3, FDIFF_1:20;
A6:
( (#Z 2) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * arccot) `| Z) . x = - ((2 * ((arccot . x) #Z (2 - 1))) / (1 + (x ^2))) ) )
by A1, SIN_COS9:92;
for x being Real st x in Z holds
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = (arccot . x) / (1 + (x ^2))
proof
let x be
Real;
( x in Z implies ((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = (arccot . x) / (1 + (x ^2)) )
assume A7:
x in Z
;
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = (arccot . x) / (1 + (x ^2))
then A8:
(1 / 2) (#) ((#Z 2) * arccot) is_differentiable_in x
by A4, FDIFF_1:9;
A9:
(#Z 2) * arccot is_differentiable_in x
by A6, A7, FDIFF_1:9;
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x =
diff (
(- ((1 / 2) (#) ((#Z 2) * arccot))),
x)
by A5, A7, FDIFF_1:def 7
.=
(- 1) * (diff (((1 / 2) (#) ((#Z 2) * arccot)),x))
by A8, FDIFF_1:15
.=
(- 1) * ((1 / 2) * (diff (((#Z 2) * arccot),x)))
by A9, FDIFF_1:15
.=
(- 1) * ((1 / 2) * ((((#Z 2) * arccot) `| Z) . x))
by A6, A7, FDIFF_1:def 7
.=
(- 1) * ((1 / 2) * (- ((2 * ((arccot . x) #Z (2 - 1))) / (1 + (x ^2)))))
by A1, A7, SIN_COS9:92
.=
(- 1) * (- ((1 / 2) * ((2 * ((arccot . x) #Z 1)) / (1 + (x ^2)))))
.=
(- 1) * (- ((1 / 2) * ((2 * (arccot . x)) / (1 + (x ^2)))))
by PREPOWER:35
.=
(arccot . x) / (1 + (x ^2))
;
hence
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = (arccot . x) / (1 + (x ^2))
;
verum
end;
hence
( - ((1 / 2) (#) ((#Z 2) * arccot)) is_differentiable_on Z & ( for x being Real st x in Z holds
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = (arccot . x) / (1 + (x ^2)) ) )
by A3, A4, FDIFF_1:20; verum