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