let Z be open Subset of REAL ; :: thesis:
let f, f1 be PartFunc of REAL ,REAL ; :: thesis:
assume that
A1: ( Z c= dom f & f = ln * (exp_R / ((#Z 2) * (f1 - exp_R ))) ) and
A2: for x being Real st x in Z holds
( f1 . x = 1 & (f1 - exp_R ) . x > 0 ) ; :: thesis:
A3: for x being Real st x in Z holds
f1 . x = 1 by A2;
for y being set st y in Z holds
y in dom (exp_R / ((#Z 2) * (f1 - exp_R ))) by A1, 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 & ( for x being Real st x in Z holds
(((#Z 2) * (f1 - exp_R )) `| Z) . x = - ((2 * (exp_R . x)) * (1 - (exp_R . x))) ) ) by A3, Th31;
A7: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
A8: for x being Real st x in Z holds
((#Z 2) * (f1 - exp_R )) . x > 0

proof

let x be Real; :: thesis:
assume A9: x in Z ; :: thesis:
then A10: ((#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 - exp_R ) . x > 0 by A2, A9;
hence ((#Z 2) * (f1 - exp_R )) . x > 0 by A10, PREPOWER:49; :: thesis:

end;

then for x being Real st x in Z holds
((#Z 2) * (f1 - exp_R )) . x <> 0 ;
then A11: exp_R / ((#Z 2) * (f1 - exp_R )) is_differentiable_on Z by A6, A7, FDIFF_2:21;
for y being set st y in Z holds
y in dom (f1 - exp_R ) by A5, FUNCT_1:21;
then A12: Z c= dom (f1 - exp_R ) by TARSKI:def 3;
A13: for x being Real st x in Z holds
(exp_R / ((#Z 2) * (f1 - exp_R ))) . x > 0

proof

let x be Real; :: thesis:
assume A14: x in Z ; :: thesis:
then A15: (exp_R / ((#Z 2) * (f1 - exp_R ))) . x = (exp_R . x) * ((((#Z 2) * (f1 - exp_R )) . x) " ) by A4, 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 ;
A16: ((#Z 2) * (f1 - exp_R )) . x > 0 by A8, A14;
exp_R . x > 0 by SIN_COS:59;
hence (exp_R / ((#Z 2) * (f1 - exp_R ))) . x > 0 by A15, A16, XREAL_1:141; :: thesis:

end;

A17: 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:
assume A18: x in Z ; :: thesis:
then A19: exp_R / ((#Z 2) * (f1 - exp_R )) is_differentiable_in x by A11, FDIFF_1:16;
(exp_R / ((#Z 2) * (f1 - exp_R ))) . x > 0 by A13, A18;
hence ln * (exp_R / ((#Z 2) * (f1 - exp_R ))) is_differentiable_in x by A19, TAYLOR_1:20; :: thesis:

end;

then A20: f is_differentiable_on Z by A1, 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:
assume A21: x in Z ; :: thesis:
then A22: ((#Z 2) * (f1 - exp_R )) . x <> 0 by A8;
A23: (#Z 2) * (f1 - exp_R ) is_differentiable_in x by A6, A21, FDIFF_1:16;
A24: exp_R is_differentiable_in x by SIN_COS:70;
A25: exp_R / ((#Z 2) * (f1 - exp_R )) is_differentiable_in x by A11, A21, FDIFF_1:16;
A26: (exp_R / ((#Z 2) * (f1 - exp_R ))) . x > 0 by A13, A21;
A27: (f1 - exp_R ) . x > 0 by A2, A21;
A28: (f1 - exp_R ) . x = (f1 . x) - (exp_R . x) by A12, A21, VALUED_1:13
.= 1 - (exp_R . x) by A2, A21 ;
then 1 - (exp_R . x) > 0 by A2, A21;
then A29: (1 - (exp_R . x)) #Z 2 > 0 by PREPOWER:49;
exp_R . x > 0 by SIN_COS:59;
then A30: (exp_R . x) / ((1 - (exp_R . x)) #Z 2) <> 0 by A29, XREAL_1:141;
A31: ((#Z 2) * (f1 - exp_R )) . x = (#Z 2) . ((f1 - exp_R ) . x) by A5, A21, FUNCT_1:22
.= ((f1 - exp_R ) . x) #Z 2 by TAYLOR_1:def 1
.= ((f1 . x) - (exp_R . x)) #Z 2 by A12, A21, VALUED_1:13
.= (1 - (exp_R . x)) #Z 2 by A2, A21 ;
A32: 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 A22, A23, A24, 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, A21, 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, A21, A31, Th31
.= ((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 ;
A33: (exp_R / ((#Z 2) * (f1 - exp_R ))) . x = (exp_R . x) * ((((#Z 2) * (f1 - exp_R )) . x) " ) by A4, A21, 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 A31, XCMPLX_1:100 ;
A34: (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 A27, A28, 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 A1, A20, A21, 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, A26, A32, A33, 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 A30, A34, XCMPLX_1:92
.= (1 + (exp_R . x)) / (1 - (exp_R . x)) by A27, A28, XCMPLX_1:92 ;
hence (f `| Z) . x = (1 + (exp_R . x)) / (1 - (exp_R . x)) ; :: thesis:

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, A17, FDIFF_1:16; :: thesis: