let r be Real; for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arccot * 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) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) )
let Z be open Subset of REAL; for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arccot * 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) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) )
let f, f1, f2 be PartFunc of REAL,REAL; ( Z c= dom (((id Z) (#) (arccot * 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) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) ) )
assume that
A1:
Z c= dom (((id Z) (#) (arccot * 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
and
A6:
for x being Real st x in Z holds
f . x = x / r
; ( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) )
A7:
Z c= (dom ((id Z) (#) (arccot * f))) /\ (dom ((r / 2) (#) (ln * (f1 + f2))))
by A1, VALUED_1:def 1;
then A8:
Z c= dom ((r / 2) (#) (ln * (f1 + f2)))
by XBOOLE_1:18;
then A9:
(r / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z
by A2, A4, A5, A6, Th108;
A10:
Z c= dom ((id Z) (#) (arccot * f))
by A7, XBOOLE_1:18;
then A11:
(id Z) (#) (arccot * f) is_differentiable_on Z
by A3, Th106;
for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r)
proof
let x be
Real;
( x in Z implies ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) )
assume A12:
x in Z
;
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r)
hence ((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x =
(diff (((id Z) (#) (arccot * f)),x)) + (diff (((r / 2) (#) (ln * (f1 + f2))),x))
by A1, A11, A9, FDIFF_1:18
.=
((((id Z) (#) (arccot * f)) `| Z) . x) + (diff (((r / 2) (#) (ln * (f1 + f2))),x))
by A11, A12, FDIFF_1:def 7
.=
((((id Z) (#) (arccot * f)) `| Z) . x) + ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x)
by A9, A12, FDIFF_1:def 7
.=
((arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))) + ((((r / 2) (#) (ln * (f1 + f2))) `| Z) . x)
by A3, A10, A12, Th106
.=
((arccot . (x / r)) - (x / (r * (1 + ((x / r) ^2))))) + (x / (r * (1 + ((x / r) ^2))))
by A2, A4, A5, A6, A8, A12, Th108
.=
arccot . (x / r)
;
verum
end;
hence
( ((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccot * f)) + ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arccot . (x / r) ) )
by A1, A11, A9, FDIFF_1:18; verum