let Z be open Subset of REAL ; :: thesis: for f, f1 being PartFunc of REAL ,REAL st Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 + exp_R ))) & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ) )
let f, f1 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 + exp_R ))) & ( for x being Real st x in Z holds
f1 . x = 1 ) implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ) ) )
assume that
A1: Z c= dom f and
A2: f = ln * (exp_R / ((#Z 2) * (f1 + exp_R ))) and
A3: for x being Real st x in Z holds
f1 . x = 1 ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ) )
for y being set st y in Z holds
y in dom (exp_R / ((#Z 2) * (f1 + exp_R ))) by A1, A2, FUNCT_1:21;
then A4: Z c= dom (exp_R / ((#Z 2) * (f1 + exp_R ))) by TARSKI:def 3;
then Z c= (dom exp_R ) /\ ((dom ((#Z 2) * (f1 + exp_R ))) \ (((#Z 2) * (f1 + exp_R )) " {0 })) by RFUNCT_1:def 4;
then A5: Z c= dom ((#Z 2) * (f1 + exp_R )) by XBOOLE_1:1;
then A6: (#Z 2) * (f1 + exp_R ) is_differentiable_on Z by A3, Th29;
for y being set st y in Z holds
y in dom (f1 + exp_R ) by A5, FUNCT_1:21;
then A7: Z c= dom (f1 + exp_R ) by TARSKI:def 3;
A8: for x being Real st x in Z holds
((#Z 2) * (f1 + exp_R )) . x > 0

proof

let x be Real; :: thesis: ( x in Z implies ((#Z 2) * (f1 + exp_R )) . x > 0 )
assume A9: x in Z ; :: thesis: ((#Z 2) * (f1 + exp_R )) . x > 0
then (f1 + exp_R ) . x = (f1 . x) + (exp_R . x) by A7, VALUED_1:def 1
.= 1 + (exp_R . x) by A3, A9 ;
then A10: (f1 + exp_R ) . x > 0 by SIN_COS:59, XREAL_1:36;
((#Z 2) * (f1 + exp_R )) . x = (#Z 2) . ((f1 + exp_R ) . x) by A5, A9, FUNCT_1:22
.= ((f1 + exp_R ) . x) #Z 2 by TAYLOR_1:def 1 ;
hence ((#Z 2) * (f1 + exp_R )) . x > 0 by A10, PREPOWER:49; :: thesis: verum

end;

A11: for x being Real st x in Z holds
(exp_R / ((#Z 2) * (f1 + exp_R ))) . x > 0

proof

let x be Real; :: thesis: ( x in Z implies (exp_R / ((#Z 2) * (f1 + exp_R ))) . x > 0 )
A12: exp_R . x > 0 by SIN_COS:59;
assume A13: x in Z ; :: thesis: (exp_R / ((#Z 2) * (f1 + exp_R ))) . x > 0
then A14: ((#Z 2) * (f1 + exp_R )) . x > 0 by A8;
(exp_R / ((#Z 2) * (f1 + exp_R ))) . x = (exp_R . x) * ((((#Z 2) * (f1 + exp_R )) . x) " ) by A4, A13, RFUNCT_1:def 4
.= (exp_R . x) * (1 / (((#Z 2) * (f1 + exp_R )) . x)) by XCMPLX_1:217
.= (exp_R . x) / (((#Z 2) * (f1 + exp_R )) . x) by XCMPLX_1:100 ;
hence (exp_R / ((#Z 2) * (f1 + exp_R ))) . x > 0 by A14, A12, XREAL_1:141; :: thesis: verum

end;

( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((#Z 2) * (f1 + exp_R )) . x <> 0 ) ) by A8, FDIFF_1:34, TAYLOR_1:16;
then A15: exp_R / ((#Z 2) * (f1 + exp_R )) is_differentiable_on Z by A6, FDIFF_2:21;
A16: for x being Real st x in Z holds
ln * (exp_R / ((#Z 2) * (f1 + exp_R ))) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * (exp_R / ((#Z 2) * (f1 + exp_R ))) is_differentiable_in x )
assume x in Z ; :: thesis: ln * (exp_R / ((#Z 2) * (f1 + exp_R ))) is_differentiable_in x
then ( exp_R / ((#Z 2) * (f1 + exp_R )) is_differentiable_in x & (exp_R / ((#Z 2) * (f1 + exp_R ))) . x > 0 ) by A15, A11, FDIFF_1:16;
hence ln * (exp_R / ((#Z 2) * (f1 + exp_R ))) is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;

then A17: f is_differentiable_on Z by A1, A2, FDIFF_1:16;
for x being Real st x in Z holds
(f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x))

proof

let x be Real; :: thesis: ( x in Z implies (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) )
A18: exp_R is_differentiable_in x by SIN_COS:70;
assume A19: x in Z ; :: thesis: (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x))
then A20: ((#Z 2) * (f1 + exp_R )) . x = (#Z 2) . ((f1 + exp_R ) . x) by A5, FUNCT_1:22
.= ((f1 + exp_R ) . x) #Z 2 by TAYLOR_1:def 1
.= ((f1 . x) + (exp_R . x)) #Z 2 by A7, A19, VALUED_1:def 1
.= (1 + (exp_R . x)) #Z 2 by A3, A19 ;
( ((#Z 2) * (f1 + exp_R )) . x <> 0 & (#Z 2) * (f1 + exp_R ) is_differentiable_in x ) by A6, A8, A19, FDIFF_1:16;
then A21: diff (exp_R / ((#Z 2) * (f1 + exp_R ))),x = (((diff exp_R ,x) * (((#Z 2) * (f1 + exp_R )) . x)) - ((diff ((#Z 2) * (f1 + exp_R )),x) * (exp_R . x))) / ((((#Z 2) * (f1 + exp_R )) . x) ^2 ) by A18, FDIFF_2:14
.= (((exp_R . x) * (((#Z 2) * (f1 + exp_R )) . x)) - ((diff ((#Z 2) * (f1 + exp_R )),x) * (exp_R . x))) / ((((#Z 2) * (f1 + exp_R )) . x) ^2 ) by SIN_COS:70
.= (((exp_R . x) * (((#Z 2) * (f1 + exp_R )) . x)) - (((((#Z 2) * (f1 + exp_R )) `| Z) . x) * (exp_R . x))) / ((((#Z 2) * (f1 + exp_R )) . x) ^2 ) by A6, A19, FDIFF_1:def 8
.= (((exp_R . x) * ((1 + (exp_R . x)) #Z 2)) - (((2 * (exp_R . x)) * (1 + (exp_R . x))) * (exp_R . x))) / (((1 + (exp_R . x)) #Z 2) ^2 ) by A3, A5, A19, A20, Th29
.= ((exp_R . x) * (((1 + (exp_R . x)) #Z 2) - ((2 * (1 + (exp_R . x))) * (exp_R . x)))) / (((1 + (exp_R . x)) #Z 2) * ((1 + (exp_R . x)) #Z 2))
.= (((exp_R . x) / ((1 + (exp_R . x)) #Z 2)) * (((1 + (exp_R . x)) #Z 2) - ((2 * (1 + (exp_R . x))) * (exp_R . x)))) / ((1 + (exp_R . x)) #Z 2) by XCMPLX_1:84 ;
A22: 1 + (exp_R . x) > 0 by SIN_COS:59, XREAL_1:36;
then ( exp_R . x > 0 & (1 + (exp_R . x)) #Z 2 > 0 ) by PREPOWER:49, SIN_COS:59;
then A23: (exp_R . x) / ((1 + (exp_R . x)) #Z 2) <> 0 by XREAL_1:141;
A24: (exp_R / ((#Z 2) * (f1 + exp_R ))) . x = (exp_R . x) * ((((#Z 2) * (f1 + exp_R )) . x) " ) by A4, A19, RFUNCT_1:def 4
.= (exp_R . x) * (1 / (((#Z 2) * (f1 + exp_R )) . x)) by XCMPLX_1:217
.= (exp_R . x) / ((1 + (exp_R . x)) #Z 2) by A20, XCMPLX_1:100 ;
A25: ( exp_R / ((#Z 2) * (f1 + exp_R )) is_differentiable_in x & (exp_R / ((#Z 2) * (f1 + exp_R ))) . x > 0 ) by A15, A11, A19, FDIFF_1:16;
A26: (1 + (exp_R . x)) #Z 2 = (1 + (exp_R . x)) #Z (1 + 1)
.= ((1 + (exp_R . x)) #Z 1) * ((1 + (exp_R . x)) #Z 1) by A22, PREPOWER:54
.= (1 + (exp_R . x)) * ((1 + (exp_R . x)) #Z 1) by PREPOWER:45
.= (1 + (exp_R . x)) * (1 + (exp_R . x)) by PREPOWER:45 ;
(f `| Z) . x = diff (ln * (exp_R / ((#Z 2) * (f1 + exp_R )))),x by A2, A17, A19, FDIFF_1:def 8
.= ((((exp_R . x) / ((1 + (exp_R . x)) #Z 2)) * (((1 + (exp_R . x)) #Z 2) - ((2 * (1 + (exp_R . x))) * (exp_R . x)))) / ((1 + (exp_R . x)) #Z 2)) / ((exp_R . x) / ((1 + (exp_R . x)) #Z 2)) by A25, A21, A24, TAYLOR_1:20
.= (((exp_R . x) / ((1 + (exp_R . x)) #Z 2)) * (((1 + (exp_R . x)) #Z 2) - ((2 * (1 + (exp_R . x))) * (exp_R . x)))) / (((exp_R . x) / ((1 + (exp_R . x)) #Z 2)) * ((1 + (exp_R . x)) #Z 2)) by XCMPLX_1:79
.= ((1 + (exp_R . x)) * (1 - (exp_R . x))) / ((1 + (exp_R . x)) * (1 + (exp_R . x))) by A23, A26, XCMPLX_1:92
.= (1 - (exp_R . x)) / (1 + (exp_R . x)) by A22, XCMPLX_1:92 ;
hence (f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ; :: thesis: verum

end;

hence ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = (1 - (exp_R . x)) / (1 + (exp_R . x)) ) ) by A1, A2, A16, FDIFF_1:16; :: thesis: verum