let r, s be Real; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (arccot * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds
( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) )
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom (arccot * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) holds
( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom (arccot * f) & ( for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 ) ) implies ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) ) )
assume that
A1:
Z c= dom (arccot * f)
and
A2:
for x being Real st x in Z holds
( f . x = (r * x) + s & f . x > - 1 & f . x < 1 )
; ( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) )
for y being object st y in Z holds
y in dom f
by A1, FUNCT_1:11;
then A3:
Z c= dom f
;
A4:
for x being Real st x in Z holds
f . x = (r * x) + s
by A2;
then A5:
f is_differentiable_on Z
by A3, FDIFF_1:23;
A6:
for x being Real st x in Z holds
arccot * f is_differentiable_in x
then A10:
arccot * f is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2)))
proof
let x be
Real;
( x in Z implies ((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) )
assume A11:
x in Z
;
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2)))
then A12:
f . x > - 1
by A2;
A13:
f . x < 1
by A2, A11;
f is_differentiable_in x
by A5, A11, FDIFF_1:9;
then diff (
(arccot * f),
x) =
- ((diff (f,x)) / (1 + ((f . x) ^2)))
by A12, A13, Th86
.=
- (((f `| Z) . x) / (1 + ((f . x) ^2)))
by A5, A11, FDIFF_1:def 7
.=
- (r / (1 + ((f . x) ^2)))
by A4, A3, A11, FDIFF_1:23
.=
- (r / (1 + (((r * x) + s) ^2)))
by A2, A11
;
hence
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2)))
by A10, A11, FDIFF_1:def 7;
verum
end;
hence
( arccot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * f) `| Z) . x = - (r / (1 + (((r * x) + s) ^2))) ) )
by A1, A6, FDIFF_1:9; verum