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∈
∈
=
a
k
i
,
l
j
.
where for each
k
i
M
and each
l
j
N
,
b
k
i
,
l
j
⊆
⊆
⊆
⊆
Obviously, for every IM
A
and sets
M
1
M
2
K
and
N
1
N
2
L
the
equality
pr
M
1
,
N
1
pr
M
2
,
N
2
A
=
pr
M
1
,
N
1
A
holds.
Theorem 4
Fo r M
⊆
K
,
N
⊆
L, the equalities
pr
M
,
N
A
=
A
)
,
(
K
−
M
,
L
−
N
A
M
,
N
=
pr
K
−
M
,
L
−
N
A
hold.
1.8 Operation “Substitution” Over an
R
-IM,
(
0
,
1
)
-IM
and L-IM
Let IM
A
be given.
First, local substitution over the IM is defined for the couples of indices
=[
K
,
L
,
{
a
k
,
l
}]
(
,
)
p
k
(
,
)
and/or
q
l
, respectively, by
p
k
;⊥
A
=
(
a
k
,
l
}
,
K
−{
k
}
)
∪{
p
}
,
L
,
{
⊥;
A
=
K
a
k
,
l
}
,
q
l
,(
L
−{
l
}
)
∪{
q
}
,
{
Second,
p
k
;
A
p
k
;⊥
⊥;
A
q
l
q
l
=
,
i.e.
p
k
;
A
=
(
a
k
,
l
}
.
q
l
K
−{
k
}
)
∪{
p
}
,(
L
−{
l
}
)
∪{
q
}
,
{
Obviously, for the above indices
k
,
l
,
p
,
q
:
k
p
;⊥
l
q
;⊥
⊥;
A
A
p
k
q
l
(
)
=
(
⊥;
),
k
p
;
p
k
;
A
l
q
q
l
(
)
=
.
A
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