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 & ( 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 ) )
A4: Z c= (dom ((id Z) (#) arctan )) /\ (dom ((1 / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:12;
A6: Z c= dom ((1 / 2) (#) (ln * (f1 + f2))) by A4, XBOOLE_1:18;
A7: ( (id Z) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arctan ) `| Z) . x = (arctan . x) + (x / (1 + (x ^2 ))) ) ) by A2, Th93;
A8: ( (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 A3, A6, Th100;
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 A9: 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, A8, FDIFF_1:27
.= ((((id Z) (#) arctan ) `| Z) . x) - (diff ((1 / 2) (#) (ln * (f1 + f2))),x) by A7, A9, FDIFF_1:def 8
.= ((((id Z) (#) arctan ) `| Z) . x) - ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A8, A9, FDIFF_1:def 8
.= ((arctan . x) + (x / (1 + (x ^2 )))) - ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A2, A9, Th93
.= ((arctan . x) + (x / (1 + (x ^2 )))) - (x / (1 + (x ^2 ))) by A3, A6, A9, Th100
.= 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, A8, FDIFF_1:27; :: thesis: verum