let a be Real; for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom ((id Z) - (a (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) holds
( (id Z) - (a (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x) ) )
let Z be open Subset of REAL; for f, f1 being PartFunc of REAL,REAL st Z c= dom ((id Z) - (a (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) holds
( (id Z) - (a (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x) ) )
let f, f1 be PartFunc of REAL,REAL; ( Z c= dom ((id Z) - (a (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) implies ( (id Z) - (a (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x) ) ) )
assume that
A1:
Z c= dom ((id Z) - (a (#) f))
and
A2:
f = ln * f1
and
A3:
for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 )
; ( (id Z) - (a (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x) ) )
A4:
Z c= (dom (id Z)) /\ (dom (a (#) f))
by A1, VALUED_1:12;
then A5:
Z c= dom (a (#) f)
by XBOOLE_1:18;
then A6:
Z c= dom (ln * f1)
by A2, VALUED_1:def 5;
then A7:
f is_differentiable_on Z
by A2, A3, Th1;
then A8:
a (#) f is_differentiable_on Z
by A5, FDIFF_1:20;
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 A4, 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
((a (#) f) `| Z) . x = a / (a + x)
for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x)
proof
let x be
Real;
( x in Z implies (((id Z) - (a (#) f)) `| Z) . x = x / (a + x) )
assume A14:
x in Z
;
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x)
then A15:
(
f1 . x = a + x &
f1 . x > 0 )
by A3;
(((id Z) - (a (#) f)) `| Z) . x =
(diff ((id Z),x)) - (diff ((a (#) f),x))
by A1, A11, A8, A14, FDIFF_1:19
.=
(((id Z) `| Z) . x) - (diff ((a (#) f),x))
by A11, A14, FDIFF_1:def 7
.=
(((id Z) `| Z) . x) - (((a (#) f) `| Z) . x)
by A8, A14, FDIFF_1:def 7
.=
1
- (((a (#) f) `| Z) . x)
by A10, A9, A14, FDIFF_1:23
.=
1
- (a / (a + x))
by A12, A14
.=
((1 * (a + x)) - a) / (a + x)
by A15, XCMPLX_1:127
.=
x / (a + x)
;
hence
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x)
;
verum
end;
hence
( (id Z) - (a (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - (a (#) f)) `| Z) . x = x / (a + x) ) )
by A1, A11, A8, FDIFF_1:19; verum