let Z be open Subset of REAL; :: thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) & Z c= ].(- 1),1.[ & f2 = #Z 2 & ( for x being Real st x in Z holds
f1 . x = 1 ) implies ( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) ) )
assume that
A1: Z c= dom (((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) and
A2: Z c= ].(- 1),1.[ and
A3: f2 = #Z 2 and
A4: for x being Real st x in Z holds
f1 . x = 1 ; :: thesis: ( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) )
Z c= (dom ((id Z) (#) arctan)) /\ (dom ((1 / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:12;
then A5: Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) by XBOOLE_1:18;
then A6: (1 / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z by A3, A4, Th102;
A7: (id Z) (#) arctan is_differentiable_on Z by A2, Th95;
for x being Real st x in Z holds
((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x

proof

let x be Real; :: thesis: ( x in Z implies ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x )
assume A8: x in Z ; :: thesis: ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x
hence ((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = (diff (((id Z) (#) arctan),x)) - (diff (((1 / 2) (#) (ln * (f1 + f2))),x)) by A1, A7, A6, FDIFF_1:19
.= ((((id Z) (#) arctan) `| Z) . x) - (diff (((1 / 2) (#) (ln * (f1 + f2))),x)) by A7, A8, FDIFF_1:def 7
.= ((((id Z) (#) arctan) `| Z) . x) - ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A6, A8, FDIFF_1:def 7
.= ((arctan . x) + (x / (1 + (x ^2)))) - ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A2, A8, Th95
.= ((arctan . x) + (x / (1 + (x ^2)))) - (x / (1 + (x ^2))) by A3, A4, A5, A8, Th102
.= arctan . x ;
:: thesis: verum

end;

hence ( ((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arctan) - ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . x ) ) by A1, A7, A6, FDIFF_1:19; :: thesis: verum