let a, b be Real; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (cot * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( cot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * f) `| Z) . x = - (a / ((sin . ((a * x) + b)) ^2)) ) )
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom (cot * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( cot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * f) `| Z) . x = - (a / ((sin . ((a * x) + b)) ^2)) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom (cot * f) & ( for x being Real st x in Z holds
f . x = (a * x) + b ) implies ( cot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * f) `| Z) . x = - (a / ((sin . ((a * x) + b)) ^2)) ) ) )
assume that
A1:
Z c= dom (cot * f)
and
A2:
for x being Real st x in Z holds
f . x = (a * x) + b
; ( cot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * f) `| Z) . x = - (a / ((sin . ((a * x) + b)) ^2)) ) )
dom (cot * f) c= dom f
by RELAT_1:25;
then A3:
Z c= dom f
by A1, XBOOLE_1:1;
then A4:
f is_differentiable_on Z
by A2, FDIFF_1:23;
A5:
for x being Real st x in Z holds
sin . (f . x) <> 0
A6:
for x being Real st x in Z holds
cot * f is_differentiable_in x
then A9:
cot * f is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cot * f) `| Z) . x = - (a / ((sin . ((a * x) + b)) ^2))
proof
let x be
Real;
( x in Z implies ((cot * f) `| Z) . x = - (a / ((sin . ((a * x) + b)) ^2)) )
assume A10:
x in Z
;
((cot * f) `| Z) . x = - (a / ((sin . ((a * x) + b)) ^2))
then A11:
f is_differentiable_in x
by A4, FDIFF_1:9;
A12:
sin . (f . x) <> 0
by A5, A10;
then
cot is_differentiable_in f . x
by FDIFF_7:47;
then diff (
(cot * f),
x) =
(diff (cot,(f . x))) * (diff (f,x))
by A11, FDIFF_2:13
.=
(- (1 / ((sin . (f . x)) ^2))) * (diff (f,x))
by A12, FDIFF_7:47
.=
- ((diff (f,x)) / ((sin . (f . x)) ^2))
.=
- ((diff (f,x)) / ((sin . ((a * x) + b)) ^2))
by A2, A10
.=
- (((f `| Z) . x) / ((sin . ((a * x) + b)) ^2))
by A4, A10, FDIFF_1:def 7
.=
- (a / ((sin . ((a * x) + b)) ^2))
by A2, A3, A10, FDIFF_1:23
;
hence
((cot * f) `| Z) . x = - (a / ((sin . ((a * x) + b)) ^2))
by A9, A10, FDIFF_1:def 7;
verum
end;
hence
( cot * f is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * f) `| Z) . x = - (a / ((sin . ((a * x) + b)) ^2)) ) )
by A1, A6, FDIFF_1:9; verum