let a be Real; :: thesis: for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL ,REAL st Z c= dom (ln * (f1 / f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 & f2 . x > 0 & x <> 0 ) ) holds
( ln * (f1 / f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = ((2 * a) - x) / (x * (x - a)) ) )
let Z be open Subset of REAL ; :: thesis: for f1, f2 being PartFunc of REAL ,REAL st Z c= dom (ln * (f1 / f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 & f2 . x > 0 & x <> 0 ) ) holds
( ln * (f1 / f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = ((2 * a) - x) / (x * (x - a)) ) )
let f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (ln * (f1 / f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 & f2 . x > 0 & x <> 0 ) ) implies ( ln * (f1 / f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = ((2 * a) - x) / (x * (x - a)) ) ) )
assume A1:
( Z c= dom (ln * (f1 / f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 & f2 . x > 0 & x <> 0 ) ) )
; :: thesis: ( ln * (f1 / f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = ((2 * a) - x) / (x * (x - a)) ) )
then
for y being set st y in Z holds
y in dom (f1 / f2)
by FUNCT_1:21;
then
Z c= dom (f1 / f2)
by TARSKI:def 3;
then A2:
Z c= (dom f1) /\ ((dom f2) \ (f2 " {0 }))
by RFUNCT_1:def 4;
then A3:
( Z c= dom f1 & Z c= (dom f2) \ (f2 " {0 }) )
by XBOOLE_1:18;
A4:
Z c= dom f2
by A2, XBOOLE_1:1;
A5:
for x being Real st x in Z holds
f1 . x = (1 * x) + (- a)
then A7:
( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = 1 ) )
by A3, FDIFF_1:31;
A8:
for x being Real st x in Z holds
f2 is_differentiable_in x
by A1, TAYLOR_1:2;
then A9:
f2 is_differentiable_on Z
by A4, FDIFF_1:16;
A10:
for x being Real st x in Z holds
(f2 `| Z) . x = 2 * x
then A12:
( f2 is_differentiable_on Z & ( for x being Real st x in Z holds
(f2 `| Z) . x = 2 * x ) )
by A4, A8, FDIFF_1:16;
for x being Real st x in Z holds
f2 . x <> 0
by A1;
then A13:
f1 / f2 is_differentiable_on Z
by A7, A12, FDIFF_2:21;
A14:
for x being Real st x in Z holds
((f1 / f2) `| Z) . x = ((2 * a) - x) / (x |^ 3)
proof
let x be
Real;
:: thesis: ( x in Z implies ((f1 / f2) `| Z) . x = ((2 * a) - x) / (x |^ 3) )
assume A15:
x in Z
;
:: thesis: ((f1 / f2) `| Z) . x = ((2 * a) - x) / (x |^ 3)
then A16:
f1 is_differentiable_in x
by A7, FDIFF_1:16;
A17:
f2 is_differentiable_in x
by A1, TAYLOR_1:2;
A18:
x <> 0
by A1, A15;
A19:
f2 . x =
x #Z 2
by A1, TAYLOR_1:def 1
.=
x |^ 2
by PREPOWER:46
;
then
(
f2 . x <> 0 &
f1 . x = x - a &
f2 . x = x |^ 2 )
by A1, A15;
then diff (f1 / f2),
x =
(((diff f1,x) * (f2 . x)) - ((diff f2,x) * (f1 . x))) / ((f2 . x) ^2 )
by A16, A17, FDIFF_2:14
.=
((((f1 `| Z) . x) * (f2 . x)) - ((diff f2,x) * (f1 . x))) / ((f2 . x) ^2 )
by A7, A15, FDIFF_1:def 8
.=
((((f1 `| Z) . x) * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2 )
by A12, A15, FDIFF_1:def 8
.=
((1 * (f2 . x)) - (((f2 `| Z) . x) * (f1 . x))) / ((f2 . x) ^2 )
by A3, A5, A15, FDIFF_1:31
.=
((1 * (f2 . x)) - ((2 * x) * (f1 . x))) / ((f2 . x) ^2 )
by A10, A15
.=
((x |^ (1 + 1)) - ((2 * x) * (x - a))) / ((x |^ 2) ^2 )
by A1, A15, A19
.=
(((x |^ 1) * x) - ((2 * x) * (x - a))) / ((x |^ 2) ^2 )
by NEWTON:11
.=
((x * x) - ((2 * x) * (x - a))) / ((x |^ 2) ^2 )
by NEWTON:10
.=
(x * ((2 * a) - x)) / (x |^ (2 + 2))
by NEWTON:13
.=
(x * ((2 * a) - x)) / (x |^ (3 + 1))
.=
(x * ((2 * a) - x)) / ((x |^ 3) * x)
by NEWTON:11
.=
((2 * a) - x) / (x |^ 3)
by A18, XCMPLX_1:92
;
hence
((f1 / f2) `| Z) . x = ((2 * a) - x) / (x |^ 3)
by A13, A15, FDIFF_1:def 8;
:: thesis: verum
end;
A20:
for x being Real st x in Z holds
(f1 / f2) . x > 0
A24:
for x being Real st x in Z holds
ln * (f1 / f2) is_differentiable_in x
then A27:
ln * (f1 / f2) is_differentiable_on Z
by A1, FDIFF_1:16;
for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = ((2 * a) - x) / (x * (x - a))
proof
let x be
Real;
:: thesis: ( x in Z implies ((ln * (f1 / f2)) `| Z) . x = ((2 * a) - x) / (x * (x - a)) )
assume A28:
x in Z
;
:: thesis: ((ln * (f1 / f2)) `| Z) . x = ((2 * a) - x) / (x * (x - a))
then A29:
f1 / f2 is_differentiable_in x
by A13, FDIFF_1:16;
A30:
f2 . x =
x #Z 2
by A1, TAYLOR_1:def 1
.=
x |^ 2
by PREPOWER:46
;
then A31:
x |^ 2
> 0
by A1, A28;
A32:
x in dom (f1 / f2)
by A1, A28, FUNCT_1:21;
A33:
(
(f1 / f2) . x > 0 &
f1 . x = x - a &
f2 . x = x |^ 2 )
by A1, A20, A28, A30;
then diff (ln * (f1 / f2)),
x =
(diff (f1 / f2),x) / ((f1 / f2) . x)
by A29, TAYLOR_1:20
.=
(((f1 / f2) `| Z) . x) / ((f1 / f2) . x)
by A13, A28, FDIFF_1:def 8
.=
(((2 * a) - x) / (x |^ 3)) / ((f1 / f2) . x)
by A14, A28
.=
(((2 * a) - x) / (x |^ 3)) / ((f1 . x) * ((f2 . x) " ))
by A32, RFUNCT_1:def 4
.=
(((2 * a) - x) / (x |^ (2 + 1))) / ((x - a) / (x |^ 2))
by A33, XCMPLX_0:def 9
.=
(((2 * a) - x) / ((x |^ 2) * x)) / ((x - a) / (x |^ 2))
by NEWTON:11
.=
((((2 * a) - x) / (x |^ 2)) / x) / ((x - a) / (x |^ 2))
by XCMPLX_1:79
.=
((((2 * a) - x) / (x |^ 2)) / ((x - a) / (x |^ 2))) / x
by XCMPLX_1:48
.=
(((2 * a) - x) / (x - a)) / x
by A31, XCMPLX_1:55
.=
((2 * a) - x) / (x * (x - a))
by XCMPLX_1:79
;
hence
((ln * (f1 / f2)) `| Z) . x = ((2 * a) - x) / (x * (x - a))
by A27, A28, FDIFF_1:def 8;
:: thesis: verum
end;
hence
( ln * (f1 / f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 / f2)) `| Z) . x = ((2 * a) - x) / (x * (x - a)) ) )
by A1, A24, FDIFF_1:16; :: thesis: verum