let a be Real; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (((- (1 / a)) (#) (cot * f)) - (id Z)) & ( for x being Real st x in Z holds
( f . x = a * x & a <> 0 ) ) holds
( ((- (1 / a)) (#) (cot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((- (1 / a)) (#) (cot * f)) - (id Z)) `| Z) . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2) ) )
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom (((- (1 / a)) (#) (cot * f)) - (id Z)) & ( for x being Real st x in Z holds
( f . x = a * x & a <> 0 ) ) holds
( ((- (1 / a)) (#) (cot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((- (1 / a)) (#) (cot * f)) - (id Z)) `| Z) . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom (((- (1 / a)) (#) (cot * f)) - (id Z)) & ( for x being Real st x in Z holds
( f . x = a * x & a <> 0 ) ) implies ( ((- (1 / a)) (#) (cot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((- (1 / a)) (#) (cot * f)) - (id Z)) `| Z) . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2) ) ) )
assume that
A1:
Z c= dom (((- (1 / a)) (#) (cot * f)) - (id Z))
and
A2:
for x being Real st x in Z holds
( f . x = a * x & a <> 0 )
; ( ((- (1 / a)) (#) (cot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((- (1 / a)) (#) (cot * f)) - (id Z)) `| Z) . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2) ) )
A3:
Z c= (dom ((- (1 / a)) (#) (cot * f))) /\ (dom (id Z))
by A1, VALUED_1:12;
then A4:
Z c= dom ((- (1 / a)) (#) (cot * f))
by XBOOLE_1:18;
then A5:
Z c= dom (cot * f)
by VALUED_1:def 5;
A6:
for x being Real st x in Z holds
f . x = (a * x) + 0
by A2;
then A7:
cot * f is_differentiable_on Z
by A5, Th7;
then A8:
(- (1 / a)) (#) (cot * f) is_differentiable_on Z
by A4, FDIFF_1:20;
set g = (- (1 / a)) (#) (cot * f);
A9:
for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0
by FUNCT_1:18;
A10:
Z c= dom (id Z)
by A3, XBOOLE_1:18;
then A11:
id Z is_differentiable_on Z
by A9, FDIFF_1:23;
A12:
for x being Real st x in Z holds
sin . (f . x) <> 0
for x being Real st x in Z holds
((((- (1 / a)) (#) (cot * f)) - (id Z)) `| Z) . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2)
proof
let x be
Real;
( x in Z implies ((((- (1 / a)) (#) (cot * f)) - (id Z)) `| Z) . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2) )
assume A13:
x in Z
;
((((- (1 / a)) (#) (cot * f)) - (id Z)) `| Z) . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2)
then A14:
f . x = (a * x) + 0
by A2;
sin . (f . x) <> 0
by A12, A13;
then A15:
(sin . (a * x)) ^2 > 0
by A14, SQUARE_1:12;
((((- (1 / a)) (#) (cot * f)) - (id Z)) `| Z) . x =
(diff (((- (1 / a)) (#) (cot * f)),x)) - (diff ((id Z),x))
by A1, A8, A11, A13, FDIFF_1:19
.=
((((- (1 / a)) (#) (cot * f)) `| Z) . x) - (diff ((id Z),x))
by A8, A13, FDIFF_1:def 7
.=
((- (1 / a)) * (diff ((cot * f),x))) - (diff ((id Z),x))
by A4, A7, A13, FDIFF_1:20
.=
((- (1 / a)) * (((cot * f) `| Z) . x)) - (diff ((id Z),x))
by A7, A13, FDIFF_1:def 7
.=
((- (1 / a)) * (((cot * f) `| Z) . x)) - (((id Z) `| Z) . x)
by A11, A13, FDIFF_1:def 7
.=
((- (1 / a)) * (- (a / ((sin . (a * x)) ^2)))) - (((id Z) `| Z) . x)
by A5, A6, A13, A14, Th7
.=
((1 / ((sin . (a * x)) ^2)) * (a / a)) - 1
by A10, A9, A13, FDIFF_1:23
.=
((1 / ((sin . (a * x)) ^2)) * 1) - 1
by A2, A13, XCMPLX_1:60
.=
(1 / ((sin . (a * x)) ^2)) - (((sin . (a * x)) ^2) / ((sin . (a * x)) ^2))
by A15, XCMPLX_1:60
.=
(1 - ((sin . (a * x)) ^2)) / ((sin . (a * x)) ^2)
.=
((((cos . (a * x)) ^2) + ((sin . (a * x)) ^2)) - ((sin . (a * x)) ^2)) / ((sin . (a * x)) ^2)
by SIN_COS:28
.=
((cos . (a * x)) ^2) / ((sin . (a * x)) ^2)
;
hence
((((- (1 / a)) (#) (cot * f)) - (id Z)) `| Z) . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2)
;
verum
end;
hence
( ((- (1 / a)) (#) (cot * f)) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((((- (1 / a)) (#) (cot * f)) - (id Z)) `| Z) . x = ((cos . (a * x)) ^2) / ((sin . (a * x)) ^2) ) )
by A1, A8, A11, FDIFF_1:19; verum