let Z be open Subset of REAL; :: thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) implies ( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) and
A2: f2 = #Z 2 and
A3: for x being Real st x in Z holds
f1 . x = 1 ; :: thesis: ( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) )
A4: Z c= dom (ln * (f1 + f2)) by A1, VALUED_1:def 5;
then for y being object st y in Z holds
y in dom (f1 + f2) by FUNCT_1:11;
then A5: Z c= dom (f1 + f2) ;
then A6: f1 + f2 is_differentiable_on Z by A2, A3, Th101;
for x being Real st x in Z holds
ln * (f1 + f2) is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * (f1 + f2) is_differentiable_in x )
assume A7: x in Z ; :: thesis: ln * (f1 + f2) is_differentiable_in x
then (f1 + f2) . x = (f1 . x) + (f2 . x) by A5, VALUED_1:def 1
.= 1 + (f2 . x) by A3, A7
.= 1 + (x #Z (1 + 1)) by A2, TAYLOR_1:def 1
.= 1 + ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= 1 + (x * (x #Z 1)) by PREPOWER:35
.= 1 + (x * x) by PREPOWER:35 ;
then A8: (f1 + f2) . x > 0 by XREAL_1:34, XREAL_1:63;
f1 + f2 is_differentiable_in x by A6, A7, FDIFF_1:9;
hence ln * (f1 + f2) is_differentiable_in x by A8, TAYLOR_1:20; :: thesis: verum

end;

then A9: ln * (f1 + f2) is_differentiable_on Z by A4, FDIFF_1:9;
for x being Real st x in Z holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2))

proof

let x be Real; :: thesis: ( x in Z implies (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) )
assume A10: x in Z ; :: thesis: (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2))
then A11: f1 + f2 is_differentiable_in x by A6, FDIFF_1:9;
A12: (f1 + f2) . x = (f1 . x) + (f2 . x) by A5, A10, VALUED_1:def 1
.= 1 + (f2 . x) by A3, A10
.= 1 + (x #Z (1 + 1)) by A2, TAYLOR_1:def 1
.= 1 + ((x #Z 1) * (x #Z 1)) by TAYLOR_1:1
.= 1 + (x * (x #Z 1)) by PREPOWER:35
.= 1 + (x * x) by PREPOWER:35 ;
then (f1 + f2) . x > 0 by XREAL_1:34, XREAL_1:63;
then diff ((ln * (f1 + f2)),x) = (diff ((f1 + f2),x)) / ((f1 + f2) . x) by A11, TAYLOR_1:20
.= (((f1 + f2) `| Z) . x) / ((f1 + f2) . x) by A6, A10, FDIFF_1:def 7
.= (2 * x) / (1 + (x ^2)) by A2, A3, A5, A10, A12, Th101 ;
hence (((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = (1 / 2) * ((2 * x) / (1 + (x ^2))) by A1, A9, A10, FDIFF_1:20
.= x / (1 + (x ^2)) ;
:: thesis: verum

end;

hence ( (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (1 + (x ^2)) ) ) by A1, A9, FDIFF_1:20; :: thesis: verum