let Z be open Subset of REAL; :: thesis: 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 ) 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; :: thesis: ( Z c= dom (ln * (exp_R + f)) & ( for x being Real st x in Z holds

f . x = 1 ) 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 ; :: thesis: ( 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:def 1;

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_1:18;

A9: for x being Real st x in Z holds

((exp_R + f) `| Z) . x = exp_R . x

(exp_R + f) . x > 0

ln * (exp_R + f) is_differentiable_in x

for x being Real st x in Z holds

((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1)

((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) ) ) by A1, A13, FDIFF_1:9; :: thesis: verum

f . x = 1 ) 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; :: thesis: ( Z c= dom (ln * (exp_R + f)) & ( for x being Real st x in Z holds

f . x = 1 ) 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 ; :: thesis: ( 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:def 1;

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_1:18;

A9: for x being Real st x in Z holds

((exp_R + f) `| Z) . x = exp_R . x

proof

A11:
for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies ((exp_R + f) `| Z) . x = exp_R . x )

assume A10: x in Z ; :: thesis: ((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_1:18

.= (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 ;

:: thesis: verum

end;assume A10: x in Z ; :: thesis: ((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_1:18

.= (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 ;

:: thesis: verum

(exp_R + f) . x > 0

proof

A13:
for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies (exp_R + f) . x > 0 )

assume A12: x in Z ; :: thesis: (exp_R + f) . x > 0

then (exp_R + f) . x = (exp_R . x) + (f . x) by A4, VALUED_1:def 1

.= (exp_R . x) + 1 by A2, A12 ;

hence (exp_R + f) . x > 0 by SIN_COS:54, XREAL_1:34; :: thesis: verum

end;assume A12: x in Z ; :: thesis: (exp_R + f) . x > 0

then (exp_R + f) . x = (exp_R . x) + (f . x) by A4, VALUED_1:def 1

.= (exp_R . x) + 1 by A2, A12 ;

hence (exp_R + f) . x > 0 by SIN_COS:54, XREAL_1:34; :: thesis: verum

ln * (exp_R + f) is_differentiable_in x

proof

then A14:
ln * (exp_R + f) is_differentiable_on Z
by A1, FDIFF_1:9;
let x be Real; :: thesis: ( x in Z implies ln * (exp_R + f) is_differentiable_in x )

assume x in Z ; :: thesis: ln * (exp_R + f) is_differentiable_in x

then ( exp_R + f is_differentiable_in x & (exp_R + f) . x > 0 ) by A8, A11, FDIFF_1:9;

hence ln * (exp_R + f) is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;assume x in Z ; :: thesis: ln * (exp_R + f) is_differentiable_in x

then ( exp_R + f is_differentiable_in x & (exp_R + f) . x > 0 ) by A8, A11, FDIFF_1:9;

hence ln * (exp_R + f) is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

for x being Real st x in Z holds

((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1)

proof

hence
( ln * (exp_R + f) is_differentiable_on Z & ( for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) )

assume A15: x in Z ; :: thesis: ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1)

then A16: (exp_R + f) . x = (exp_R . x) + (f . x) by A4, VALUED_1:def 1

.= (exp_R . x) + 1 by A2, A15 ;

( exp_R + f is_differentiable_in x & (exp_R + f) . x > 0 ) by A8, A11, A15, 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, A15, FDIFF_1:def 7

.= (exp_R . x) / ((exp_R . x) + 1) by A9, A15, A16 ;

hence ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) by A14, A15, FDIFF_1:def 7; :: thesis: verum

end;assume A15: x in Z ; :: thesis: ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1)

then A16: (exp_R + f) . x = (exp_R . x) + (f . x) by A4, VALUED_1:def 1

.= (exp_R . x) + 1 by A2, A15 ;

( exp_R + f is_differentiable_in x & (exp_R + f) . x > 0 ) by A8, A11, A15, 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, A15, FDIFF_1:def 7

.= (exp_R . x) / ((exp_R . x) + 1) by A9, A15, A16 ;

hence ((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) by A14, A15, FDIFF_1:def 7; :: thesis: verum

((ln * (exp_R + f)) `| Z) . x = (exp_R . x) / ((exp_R . x) + 1) ) ) by A1, A13, FDIFF_1:9; :: thesis: verum