let n be Element of NAT ; for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * (arccot ^))) & Z c= ].(- 1),1.[ & n > 0 holds
( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) )
let Z be open Subset of REAL; ( Z c= dom ((1 / n) (#) ((#Z n) * (arccot ^))) & Z c= ].(- 1),1.[ & n > 0 implies ( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) ) )
assume that
A1:
Z c= dom ((1 / n) (#) ((#Z n) * (arccot ^)))
and
A2:
Z c= ].(- 1),1.[
and
A3:
n > 0
; ( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) )
A4:
Z c= dom ((#Z n) * (arccot ^))
by A1, VALUED_1:def 5;
A5:
for x being Real st x in Z holds
arccot . x <> 0
proof
PI in ].0,4.[
by SIN_COS:def 28;
then
PI > 0
by XXREAL_1:4;
then A6:
PI / 4
> 0 / 4
by XREAL_1:74;
let x be
Real;
( x in Z implies arccot . x <> 0 )
assume A7:
x in Z
;
arccot . x <> 0
assume A8:
arccot . x = 0
;
contradiction
].(- 1),1.[ c= [.(- 1),1.]
by XXREAL_1:25;
then
Z c= [.(- 1),1.]
by A2, XBOOLE_1:1;
then
x in [.(- 1),1.]
by A7;
then
0 in arccot .: [.(- 1),1.]
by A8, FUNCT_1:def 6, SIN_COS9:24;
then
0 in [.(PI / 4),((3 / 4) * PI).]
by RELAT_1:115, SIN_COS9:56;
hence
contradiction
by A6, XXREAL_1:1;
verum
end;
A9:
arccot ^ is_differentiable_on Z
by A2, Th68;
for x being Real st x in Z holds
(#Z n) * (arccot ^) is_differentiable_in x
then A10:
(#Z n) * (arccot ^) is_differentiable_on Z
by A4, FDIFF_1:9;
for y being object st y in Z holds
y in dom (arccot ^)
by A4, FUNCT_1:11;
then A11:
Z c= dom (arccot ^)
by TARSKI:def 3;
for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2)))
proof
let x be
Real;
( x in Z implies (((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) )
assume A12:
x in Z
;
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2)))
then A13:
arccot ^ is_differentiable_in x
by A9, FDIFF_1:9;
A14:
(arccot ^) . x = 1
/ (arccot . x)
by A11, A12, RFUNCT_1:def 2;
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x =
(1 / n) * (diff (((#Z n) * (arccot ^)),x))
by A1, A10, A12, FDIFF_1:20
.=
(1 / n) * ((n * (((arccot ^) . x) #Z (n - 1))) * (diff ((arccot ^),x)))
by A13, TAYLOR_1:3
.=
(1 / n) * ((n * (((arccot ^) . x) #Z (n - 1))) * (((arccot ^) `| Z) . x))
by A9, A12, FDIFF_1:def 7
.=
(1 / n) * ((n * (((arccot ^) . x) #Z (n - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2)))))
by A2, A12, Th68
.=
(((1 / n) * n) * (((arccot ^) . x) #Z (n - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2))))
.=
(1 * (((arccot ^) . x) #Z (n - 1))) * (1 / (((arccot . x) ^2) * (1 + (x ^2))))
by A3, XCMPLX_1:106
.=
((1 / (arccot . x)) #Z (n - 1)) * (1 / (((arccot . x) #Z 2) * (1 + (x ^2))))
by A14, FDIFF_7:1
.=
(1 / ((arccot . x) #Z (n - 1))) / (((arccot . x) #Z 2) * (1 + (x ^2)))
by PREPOWER:42
.=
1
/ (((arccot . x) #Z (n - 1)) * (((arccot . x) #Z 2) * (1 + (x ^2))))
by XCMPLX_1:78
.=
1
/ ((((arccot . x) #Z (n - 1)) * ((arccot . x) #Z 2)) * (1 + (x ^2)))
.=
1
/ (((arccot . x) #Z ((n - 1) + 2)) * (1 + (x ^2)))
by A5, A12, PREPOWER:44
.=
1
/ (((arccot . x) #Z (n + 1)) * (1 + (x ^2)))
;
hence
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1
/ (((arccot . x) #Z (n + 1)) * (1 + (x ^2)))
;
verum
end;
hence
( (1 / n) (#) ((#Z n) * (arccot ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (arccot ^))) `| Z) . x = 1 / (((arccot . x) #Z (n + 1)) * (1 + (x ^2))) ) )
by A1, A10, FDIFF_1:20; verum