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={
p
1
,
p
2
,...,
p
u
}
,
={
q
1
,
q
2
,...,
q
v
}
Let the sets of indices
P
Q
be given.
Third, for them we define sequentially:
P
K
;⊥
A
p
1
k
1
A
p
2
k
2
...
p
u
k
u
;⊥
=
,
A
A
Q
L
q
1
l
1
q
2
l
2
...
q
v
⊥;
=
(
⊥;
),
l
v
P
K
;
A
P
K
;⊥
A
Q
L
Q
L
=
(
⊥;
),
Obviously, for the sets
K
,
L
,
P
,
Q
:
K
P
;⊥
P
K
;⊥
A
A
K
P
;
P
K
;
A
L
Q
Q
L
L
Q
Q
L
(
)
=
⊥;
(
⊥;
)
=
(
)
=
A
.
Theorem 5
For every four sets of indices P
1
,
P
2
,
Q
1
,
Q
2
P
2
P
1
;
P
1
K
;
A
P
2
K
;
A
Q
2
Q
1
Q
1
L
Q
2
L
=
.
1.9 An Example from Number Theory
It is well-known (see e.g., [43]) that each natural number
n
has a canonical rep-
i
=
1
k
p
α
i
i
resentation
m
=
, where
k
,
α
1
,α
2
,...,α
k
≥
1 are natural numbers and
p
1
,
p
2
,...,
p
k
are different prime numbers. We can always suppose that
p
1
<
p
2
<
···
<
p
k
.
Now, we see that
m
has the following IM-interpretation:
p
1
p
2
...
p
k
IM
(
m
,
a
)
=
α
k
,
a
α
1
α
2
...
where “
a
” is an arbitrary symbol, in a particular case—the same “
m
”.
Obviously, if
m
is a prime number, its IM-interpretation is
p
a
1
IM
(
p
,
a
)
=
and when
m
=
pq
for the prime numbers
p
and
q
, its IM-interpretation is
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