let a be Real; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (- (ln * f)) & ( for x being Real st x in Z holds
( f . x = a - x & f . x > 0 ) ) holds
( - (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (ln * f)) `| Z) . x = 1 / (a - x) ) )
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom (- (ln * f)) & ( for x being Real st x in Z holds
( f . x = a - x & f . x > 0 ) ) holds
( - (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (ln * f)) `| Z) . x = 1 / (a - x) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom (- (ln * f)) & ( for x being Real st x in Z holds
( f . x = a - x & f . x > 0 ) ) implies ( - (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (ln * f)) `| Z) . x = 1 / (a - x) ) ) )
assume that
A1:
Z c= dom (- (ln * f))
and
A2:
for x being Real st x in Z holds
( f . x = a - x & f . x > 0 )
; ( - (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (ln * f)) `| Z) . x = 1 / (a - x) ) )
then A3:
Z c= dom (ln * f)
by TARSKI:def 3;
then
for y being object st y in Z holds
y in dom f
by FUNCT_1:11;
then A4:
Z c= dom f
by TARSKI:def 3;
A5:
for x being Real st x in Z holds
f . x = ((- 1) * x) + a
proof
let x be
Real;
( x in Z implies f . x = ((- 1) * x) + a )
assume
x in Z
;
f . x = ((- 1) * x) + a
then
f . x = a - x
by A2;
hence
f . x = ((- 1) * x) + a
;
verum
end;
then A6:
f is_differentiable_on Z
by A4, FDIFF_1:23;
then A7:
ln * f is_differentiable_on Z
by A3, FDIFF_1:9;
A8:
for x being Real st x in Z holds
((- (ln * f)) `| Z) . x = 1 / (a - x)
proof
let x be
Real;
( x in Z implies ((- (ln * f)) `| Z) . x = 1 / (a - x) )
assume A9:
x in Z
;
((- (ln * f)) `| Z) . x = 1 / (a - x)
then A10:
f . x = a - x
by A2;
(
f . x > 0 &
f is_differentiable_in x )
by A2, A6, A9, FDIFF_1:9;
then diff (
(ln * f),
x) =
(diff (f,x)) / (f . x)
by TAYLOR_1:20
.=
((f `| Z) . x) / (f . x)
by A6, A9, FDIFF_1:def 7
.=
(- 1) / (a - x)
by A4, A5, A9, A10, FDIFF_1:23
;
then (((- 1) (#) (ln * f)) `| Z) . x =
(- 1) * ((- 1) / (a - x))
by A1, A7, A9, FDIFF_1:20
.=
((- 1) * (- 1)) / (a - x)
by XCMPLX_1:74
.=
1
/ (a - x)
;
hence
((- (ln * f)) `| Z) . x = 1
/ (a - x)
;
verum
end;
Z c= dom ((- 1) (#) (ln * f))
by A1;
hence
( - (ln * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((- (ln * f)) `| Z) . x = 1 / (a - x) ) )
by A7, A8, FDIFF_1:20; verum