let Z be open Subset of REAL; for f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) arccot) + ((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) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) )
let f1, f2 be PartFunc of REAL,REAL; ( Z c= dom (((id Z) (#) arccot) + ((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) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) ) )
assume that
A1:
Z c= dom (((id Z) (#) arccot) + ((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
; ( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) )
Z c= (dom ((id Z) (#) arccot)) /\ (dom ((1 / 2) (#) (ln * (f1 + f2))))
by A1, VALUED_1:def 1;
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) (#) arccot is_differentiable_on Z
by A2, Th96;
for x being Real st x in Z holds
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x
proof
let x be
Real;
( x in Z implies ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x )
assume A8:
x in Z
;
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x
hence ((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x =
(diff (((id Z) (#) arccot),x)) + (diff (((1 / 2) (#) (ln * (f1 + f2))),x))
by A1, A7, A6, FDIFF_1:18
.=
((((id Z) (#) arccot) `| Z) . x) + (diff (((1 / 2) (#) (ln * (f1 + f2))),x))
by A7, A8, FDIFF_1:def 7
.=
((((id Z) (#) arccot) `| Z) . x) + ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x)
by A6, A8, FDIFF_1:def 7
.=
((arccot . x) - (x / (1 + (x ^2)))) + ((((1 / 2) (#) (ln * (f1 + f2))) `| Z) . x)
by A2, A8, Th96
.=
((arccot . x) - (x / (1 + (x ^2)))) + (x / (1 + (x ^2)))
by A3, A4, A5, A8, Th102
.=
arccot . x
;
verum
end;
hence
( ((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccot) + ((1 / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . x ) )
by A1, A7, A6, FDIFF_1:18; verum