let Z be open Subset of REAL; ( Z c= dom (arccot * cot) & ( for x being Real st x in Z holds
( cot . x > - 1 & cot . x < 1 ) ) implies ( arccot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cot) `| Z) . x = 1 ) ) )
assume that
A1:
Z c= dom (arccot * cot)
and
A2:
for x being Real st x in Z holds
( cot . x > - 1 & cot . x < 1 )
; ( arccot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cot) `| Z) . x = 1 ) )
dom (arccot * cot) c= dom cot
by RELAT_1:25;
then A3:
Z c= dom cot
by A1, XBOOLE_1:1;
A4:
for x being Real st x in Z holds
arccot * cot is_differentiable_in x
then A7:
arccot * cot is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((arccot * cot) `| Z) . x = 1
proof
let x be
Real;
( x in Z implies ((arccot * cot) `| Z) . x = 1 )
assume A8:
x in Z
;
((arccot * cot) `| Z) . x = 1
then A9:
(
cot . x > - 1 &
cot . x < 1 )
by A2;
A10:
cot . x = (cos . x) / (sin . x)
by A3, A8, RFUNCT_1:def 1;
A11:
sin . x <> 0
by A3, A8, FDIFF_8:2;
then A12:
cot is_differentiable_in x
by FDIFF_7:47;
A13:
(sin . x) ^2 <> 0
by A11, SQUARE_1:12;
((arccot * cot) `| Z) . x =
diff (
(arccot * cot),
x)
by A7, A8, FDIFF_1:def 7
.=
- ((diff (cot,x)) / (1 + ((cot . x) ^2)))
by A12, A9, SIN_COS9:86
.=
- ((- (1 / ((sin . x) ^2))) / (1 + ((cot . x) ^2)))
by A11, FDIFF_7:47
.=
(1 / ((sin . x) ^2)) / (1 + ((cot . x) ^2))
.=
1
/ (((sin . x) ^2) * (1 + (((cos . x) / (sin . x)) * ((cos . x) / (sin . x)))))
by A10, XCMPLX_1:78
.=
1
/ (((sin . x) ^2) * (1 + (((cos . x) ^2) / ((sin . x) ^2))))
by XCMPLX_1:76
.=
1
/ (((sin . x) ^2) + ((((sin . x) ^2) * ((cos . x) ^2)) / ((sin . x) ^2)))
.=
1
/ (((sin . x) ^2) + ((cos . x) ^2))
by A13, XCMPLX_1:89
.=
1
/ 1
by SIN_COS:28
.=
1
;
hence
((arccot * cot) `| Z) . x = 1
;
verum
end;
hence
( arccot * cot is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * cot) `| Z) . x = 1 ) )
by A1, A4, FDIFF_1:9; verum