and
G
i
(
w
i
) =
{
y
i

w
i

y
i
∈ P
i
(¯
x
i
)
}
,
i
∈
N.
Denote the convex hull of the union of these sets by
G
G
=
conv
[
i
∈
N
(
G
i
(¯
x
i
)
∪ G
i
(
w
i
))
and show that
G
does not contain zero. Suppose it does. Then, by convexity of
G
i
(¯
x
i
) and
G
i
(
w
i
)
,
there exist
t
= (
t
i
)
i
∈
N
,
s
= (
s
i
)
i
∈
N
,
y, z
∈
X
such that
X
i
∈
N
t
i
+
X
i
∈
N
s
i
= 1
,
t
i
, s
i
≥
0
,
i
∈
N,
X
i
∈
N
t
i
y
i
+
X
i
∈
N
s
i
z
i
=
X
i
∈
N
t
i
¯
x
i
+
X
i
∈
N
s
i
w
i
,
(33)
where
y
i
∈ P
i
(¯
x
i
) if
i
∈
supp t,
and
z
i
∈ P
i
(¯
x
i
) if
i
∈
supp s.
This implies that
¯
x
is rejected by a fuzzy coalition
t
+
s,
a contradiction. Therefore, a zero point
does not belong to the convex set
G.
By the nonstandard separating hyperplane
theorem (see Konovalov (2001), Theorem 2.9.3), there exists
p
∈
*
IR
l
such that
for every
y
i
∈ P
i
(¯
x
i
)
, i
∈
N,
the following conditions hold simultaneously
py
i
> pw
i
,
(34)
and
py
i
> p
¯
x
i
.
(35)
Define the components of the vector
d
∈
*
IR
n
+
by
d
i
= max
{
0
, p
¯
x
i

pw
i
}
,
i
∈
N.
(36)
Then
p
¯
x
i
≤
pw
i
+
d
i
,
i
∈
N,
which implies attainability of ¯
x.
To prove the
validity of the required inclusion we have to show that ¯
x
satisfies the equilibrium
property of individual rationality:
¯
B
i
(
p, d
i
)
∩ P
i
(¯
x
i
) =
∅
.
(37)
What we do have so far is
{
x
∈
*
X
i
:
px
≤
pw
i
+
d
i
} ∩ P
i
(¯
x
i
) =
∅
,
i
∈
N.
But then (37) is a consequence of Proposition 3
.
9 and openness of the sets
P
i
(¯
x
i
)
, i
∈
N.
To prove the converse inclusion
W
ns
(
E
)
⊆
C
fr
(
E
)
,
let ¯
x
∈
W
ns
(
E
) and assume
that
p
∈
*
IR
l
and
d
∈
*
IR
n
+
are corresponding nonstandard equilibrium prices
and dividends.
Suppose that there exists a fuzzy coalition
ξ
= (
ξ
i
)
i
∈
N
that
blocks ¯
x.
Then, there exist
y
i
∈ P
i
(¯
x
i
) ,
t
i
, s
i
∈
IR
+
, i
∈
supp ξ
such that
X
i
∈
N
t
i
y
i
+
X
i
∈
N
s
i
y
i
=
X
i
∈
N
t
i
¯
x
i
+
X
i
∈
N
s
i
w
i
,
(38)
41