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1
(
)
2
~
~
∫
2
2
2
θ
=
bx
+
C
α
d
α
=
2
α
a
X
0
1
1
1
1
1
⎛
⎞
⎛
⎞
2
2
2
2
2
2
=
b
x
+
x
x
+
x
+
b
x
+
x
x
+
x
.
⎜
⎝
⎟
⎠
⎜
⎝
⎟
⎠
2
R
R
R
2
R
R
3
12
6
6
20
a
~
[
]
~
~
(
)
1
,
2
E
E
If
a
≡
b
,
b
L
b
,
is a negative number, the
α
-level set
looks like
,
α
α
R
where
[
]
−
(
)
(
)
2
E
1
=
b
x
2
2
+
2
1
−
α
x
x
+
1
−
α
x
2
α
2
R
R
[
]
(
)
(
)
(
)
2
2
2
3
2
−
b
1
−
α
x
+
2
1
−
α
x
x
+
1
−
α
x
;
L
2
R
R
[
]
+
(
)
(
)
2
2
2
1
2
E
=
b
x
−
2
1
−
α
x
x
+
1
−
α
x
α
1
L
L
[
]
(
)
(
)
(
)
2
1
2
3
2
+
b
1
−
α
x
−
2
1
−
α
x
x
+
1
−
α
x
.
R
1
L
L
Then
1
(
)
1
~
~
∫
2
2
1
θ
=
bx
+
E
α
d
α
=
2
α
a
X
0
1
1
1
1
1
⎛
⎞
⎛
⎞
2
2
2
2
2
2
=
b
x
+
x
x
+
x
−
b
x
+
x
x
+
x
;
⎜
⎝
⎟
⎠
⎜
⎝
⎟
⎠
2
R
R
L
2
R
R
3
12
6
6
20
1
(
)
2
~
~
∫
2
1
2
θ
=
bx
+
E
α
d
α
=
2
α
a
X
0
1
1
1
1
1
⎛
⎞
⎛
⎞
2
1
2
2
1
2
=
b
x
−
x
x
+
x
+
b
x
−
x
x
+
x
,
⎜
⎝
⎟
⎠
⎜
⎝
⎟
⎠
1
L
L
R
1
L
L
3
12
6
6
20
or
()
()
⎡
q
⎤
⎡
q
⎤
−
1
1
1
−
1
1
1
~
~
2
2
2
2
θ
=
b
x
+
x
x
+
x
−
b
x
+
x
x
+
x
;
⎢
⎣
⎥
⎦
⎢
⎣
⎥
⎦
2
q
q
M
M
L
q
q
M
M
a
X
3
12
6
6
20
q
q
q
q
()
()
⎡
r
⎤
⎡
r
⎤
−
1
1
1
−
1
1
2
2
2
2
2
θ
=
b
x
+
x
x
+
x
+
b
x
+
x
x
+
x
,
⎢
⎣
⎥
⎦
⎢
⎣
⎥
⎦
~
~
2
r
r
M
M
R
r
r
M
M
a
X
3
12
6
6
20
r
r
r
r
where
1
b
−
b
≥
0
⎩
⎨
⎧
⎩
⎨
⎧
L
,
q
=
1
L
q
=
M
=
q
2
b
+
b
<
0
R
,
q
=
2
R
2
b
−
b
≥
0
L
,
r
=
1
⎧
⎧
L
r
=
M
=
⎩
⎨
⎩
⎨
r
1
b
+
b
<
0
R
,
r
=
2
R
The proposition 6.5 is proved.
The Proposition 6.6.
Boundaries of the weighed segment of product of fuzzy
numbers
~
,
~
and
~
look like