let r be Real; :: thesis: for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL ,REAL st Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) )
let Z be open Subset of REAL ; :: thesis: for f, f1, f2 being PartFunc of REAL ,REAL st Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) holds
( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) )
let f, f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) & r <> 0 & ( for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) implies ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) ) )
assume that
A1: Z c= dom (((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) and
A2: r <> 0 and
A3: for x being Real st x in Z holds
( f . x = x / r & f . x > - 1 & f . x < 1 ) and
A4: for x being Real st x in Z holds
f1 . x = 1 and
A5: ( f2 = (#Z 2) * f & ( for x being Real st x in Z holds
f . x = x / r ) ) ; :: thesis: ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) )
A6: Z c= (dom ((id Z) (#) (arctan * f))) /\ (dom ((r / 2) (#) (ln * (f1 + f2)))) by A1, VALUED_1:12;
then A7: Z c= dom ((id Z) (#) (arctan * f)) by XBOOLE_1:18;
A8: Z c= dom ((r / 2) (#) (ln * (f1 + f2))) by A6, XBOOLE_1:18;
A9: ( (id Z) (#) (arctan * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arctan * f)) `| Z) . x = (arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2 )))) ) ) by A3, A7, Th103;
A10: ( (r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((r / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (r * (1 + ((x / r) ^2 ))) ) ) by A2, A4, A5, A8, Th106;
for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r)

proof

let x be Real; :: thesis: ( x in Z implies ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) )
assume A11: x in Z ; :: thesis: ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r)
hence ((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = (diff ((id Z) (#) (arctan * f)),x) - (diff ((r / 2) (#) (ln * (f1 + f2))),x) by A1, A9, A10, FDIFF_1:27
.= ((((id Z) (#) (arctan * f)) `| Z) . x) - (diff ((r / 2) (#) (ln * (f1 + f2))),x) by A9, A11, FDIFF_1:def 8
.= ((((id Z) (#) (arctan * f)) `| Z) . x) - ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A10, A11, FDIFF_1:def 8
.= ((arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2 ))))) - ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x) by A3, A7, A11, Th103
.= ((arctan . (x / r)) + (x / (r * (1 + ((x / r) ^2 ))))) - (x / (r * (1 + ((x / r) ^2 )))) by A2, A4, A5, A8, A11, Th106
.= arctan . (x / r) ;
:: thesis: verum

end;

hence ( ((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arctan * f)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) ) ) by A1, A9, A10, FDIFF_1:27; :: thesis: verum