let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom (ln * (exp_R - f)) & ( for x being Real st x in Z holds
( f . x = 1 & (exp_R - f) . x > 0 ) ) holds
( ln * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom (ln * (exp_R - f)) & ( for x being Real st x in Z holds
( f . x = 1 & (exp_R - f) . x > 0 ) ) implies ( ln * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) ) ) )
assume that
A1:
Z c= dom (ln * (exp_R - f))
and
A2:
for x being Real st x in Z holds
( f . x = 1 & (exp_R - f) . x > 0 )
; ( ln * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) ) )
A3:
for x being Real st x in Z holds
f . x = (0 * x) + 1
by A2;
for y being object st y in Z holds
y in dom (exp_R - f)
by A1, FUNCT_1:11;
then A4:
Z c= dom (exp_R - f)
by TARSKI:def 3;
then
Z c= (dom exp_R) /\ (dom f)
by VALUED_1:12;
then A5:
Z c= dom f
by XBOOLE_1:18;
then A6:
f is_differentiable_on Z
by A3, FDIFF_1:23;
A7:
exp_R is_differentiable_on Z
by FDIFF_1:26, TAYLOR_1:16;
then A8:
exp_R - f is_differentiable_on Z
by A4, A6, FDIFF_1:19;
A9:
for x being Real st x in Z holds
((exp_R - f) `| Z) . x = exp_R . x
proof
let x be
Real;
( x in Z implies ((exp_R - f) `| Z) . x = exp_R . x )
assume A10:
x in Z
;
((exp_R - f) `| Z) . x = exp_R . x
hence ((exp_R - f) `| Z) . x =
(diff (exp_R,x)) - (diff (f,x))
by A4, A6, A7, FDIFF_1:19
.=
(exp_R . x) - (diff (f,x))
by SIN_COS:65
.=
(exp_R . x) - ((f `| Z) . x)
by A6, A10, FDIFF_1:def 7
.=
(exp_R . x) - 0
by A5, A3, A10, FDIFF_1:23
.=
exp_R . x
;
verum
end;
A11:
for x being Real st x in Z holds
ln * (exp_R - f) is_differentiable_in x
then A12:
ln * (exp_R - f) is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1)
proof
let x be
Real;
( x in Z implies ((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) )
assume A13:
x in Z
;
((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1)
then A14:
(exp_R - f) . x =
(exp_R . x) - (f . x)
by A4, VALUED_1:13
.=
(exp_R . x) - 1
by A2, A13
;
(
exp_R - f is_differentiable_in x &
(exp_R - f) . x > 0 )
by A2, A8, A13, FDIFF_1:9;
then diff (
(ln * (exp_R - f)),
x) =
(diff ((exp_R - f),x)) / ((exp_R - f) . x)
by TAYLOR_1:20
.=
(((exp_R - f) `| Z) . x) / ((exp_R - f) . x)
by A8, A13, FDIFF_1:def 7
.=
(exp_R . x) / ((exp_R . x) - 1)
by A9, A13, A14
;
hence
((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1)
by A12, A13, FDIFF_1:def 7;
verum
end;
hence
( ln * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (exp_R - f)) `| Z) . x = (exp_R . x) / ((exp_R . x) - 1) ) )
by A1, A11, FDIFF_1:9; verum