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μ
A
(
x
j
)
=
λμ
B
(
x
j
), μ
A
(
x
j
)
=
λμ
B
(
v
A
(
v
B
(
x
j
),
x
j
)
=
λ
x
j
)
v
A
(
v
B
(
x
j
), π
A
(
x
j
)
=
λπ
B
(
x
j
), π
A
(
x
j
)
=
λπ
B
(
x
j
)
=
λ
x
j
)
for all
x
j
∈
X
(2.176)
while because
π
A
(
−
μ
A
(
v
A
(
x
j
), π
A
(
−
μ
A
(
v
A
(
x
j
)
=
1
x
j
)
−
x
j
)
=
1
x
j
)
−
x
j
),
for all
x
j
∈
X
(2.177)
π
B
(
−
μ
B
(
v
B
(
x
j
), π
B
(
−
μ
B
(
v
B
(
x
j
)
=
1
x
j
)
−
x
j
)
=
1
x
j
)
−
x
j
),
for all
x
j
∈
X
(2.178)
=
B
, which completes the proofs of
the conditions (1) and (2) in Definition 2.32. Furthermore, by Eq. (
2.172
), we have
A
Then by Eq. (
2.178
), we have
λ
=
1, i.e.,
,
B
x
j
j
=
1
w
j
f
A
c
8
(
A
,
B
˙
)
=
j
=
1
w
j
g
A
x
j
·
j
=
1
w
j
g
B
x
j
j
=
1
w
j
f
,
A
x
j
B
c
8
(
B
,
A
=
j
=
1
w
j
g
A
x
j
=˙
)
(2.179)
B
x
j
·
j
=
1
w
j
g
c
8
(
A
,
B
Thus, we can prove that
also satisfies the condition (3) of Definition 2.32.
If the universe of discourse,
X
, is continuous and the weight of the element
x
˙
)
∈
0 and
a
=[
,
]
(
)
(
)
≥
(
)
=
X
a
b
is
w
x
, where
w
x
w
x
dx
1, then we get the continuous
form of Eq. (
2.172
):
b
a
w
(
x
)
f
A
,
B
(
x
)
dx
c
9
(
A
,
B
˙
)
=
b
a
(2.180)
·
b
a
w
(
x
)
g
A
(
x
)
dx
w
(
x
)
g
B
(
x
)
dx
where
2
v
A
(
2
2
2
v
A
(
2
2
μ
A
(
π
A
μ
A
(
π
A
g
A
(
x
)
=
x
)
+
x
)
+
(
x
)
+
x
)
+
x
)
+
(
x
)
(2.181)
2
v
B
(
2
2
2
v
B
(
2
2
μ
B
(
π
B
μ
B
(
π
B
g
B
(
x
)
=
x
)
+
x
)
+
(
x
)
+
x
)
+
x
)
+
(
x
)
(2.182)
,
B
x
j
=
μ
A
(
) μ
B
(
v
A
(
v
B
(
)
+
π
A
) π
B
)
+
μ
A
(
) μ
B
(
f
x
x
)
+
x
)
x
(
x
(
x
x
x
)
A
v
A
(
v
B
(
)
+
π
A
) π
B
+
x
)
x
(
x
(
x
)
(2.183)
If all the elements have the same importance, then Eq. (
2.181
) reduces to
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