let a, b be Real; for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom ((id Z) - ((a + b) (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = x + b & f1 . x > 0 ) ) holds
( (id Z) - ((a + b) (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - ((a + b) (#) f)) `| Z) . x = (x - a) / (x + b) ) )
let Z be open Subset of REAL; for f, f1 being PartFunc of REAL,REAL st Z c= dom ((id Z) - ((a + b) (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = x + b & f1 . x > 0 ) ) holds
( (id Z) - ((a + b) (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - ((a + b) (#) f)) `| Z) . x = (x - a) / (x + b) ) )
let f, f1 be PartFunc of REAL,REAL; ( Z c= dom ((id Z) - ((a + b) (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = x + b & f1 . x > 0 ) ) implies ( (id Z) - ((a + b) (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - ((a + b) (#) f)) `| Z) . x = (x - a) / (x + b) ) ) )
assume that
A1:
Z c= dom ((id Z) - ((a + b) (#) f))
and
A2:
f = ln * f1
and
A3:
for x being Real st x in Z holds
( f1 . x = x + b & f1 . x > 0 )
; ( (id Z) - ((a + b) (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - ((a + b) (#) f)) `| Z) . x = (x - a) / (x + b) ) )
A4:
for x being Real st x in Z holds
( f1 . x = b + x & f1 . x > 0 )
by A3;
A5:
Z c= (dom (id Z)) /\ (dom ((a + b) (#) f))
by A1, VALUED_1:12;
then A6:
Z c= dom ((a + b) (#) f)
by XBOOLE_1:18;
then A7:
Z c= dom (ln * f1)
by A2, VALUED_1:def 5;
then A8:
f is_differentiable_on Z
by A2, A4, Th1;
then A9:
(a + b) (#) f is_differentiable_on Z
by A6, FDIFF_1:20;
A10:
for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0
by FUNCT_1:18;
A11:
Z c= dom (id Z)
by A5, XBOOLE_1:18;
then A12:
id Z is_differentiable_on Z
by A10, FDIFF_1:23;
A13:
for x being Real st x in Z holds
(((a + b) (#) f) `| Z) . x = (a + b) / (x + b)
for x being Real st x in Z holds
(((id Z) - ((a + b) (#) f)) `| Z) . x = (x - a) / (x + b)
proof
let x be
Real;
( x in Z implies (((id Z) - ((a + b) (#) f)) `| Z) . x = (x - a) / (x + b) )
assume A15:
x in Z
;
(((id Z) - ((a + b) (#) f)) `| Z) . x = (x - a) / (x + b)
then A16:
(
f1 . x = x + b &
f1 . x > 0 )
by A3;
(((id Z) - ((a + b) (#) f)) `| Z) . x =
(diff ((id Z),x)) - (diff (((a + b) (#) f),x))
by A1, A12, A9, A15, FDIFF_1:19
.=
(((id Z) `| Z) . x) - (diff (((a + b) (#) f),x))
by A12, A15, FDIFF_1:def 7
.=
(((id Z) `| Z) . x) - ((((a + b) (#) f) `| Z) . x)
by A9, A15, FDIFF_1:def 7
.=
1
- ((((a + b) (#) f) `| Z) . x)
by A11, A10, A15, FDIFF_1:23
.=
1
- ((a + b) / (x + b))
by A13, A15
.=
((1 * (x + b)) - (a + b)) / (x + b)
by A16, XCMPLX_1:127
.=
(x - a) / (x + b)
;
hence
(((id Z) - ((a + b) (#) f)) `| Z) . x = (x - a) / (x + b)
;
verum
end;
hence
( (id Z) - ((a + b) (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) - ((a + b) (#) f)) `| Z) . x = (x - a) / (x + b) ) )
by A1, A12, A9, FDIFF_1:19; verum