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the support of
ʓ
of the form
ˉ
|
Act
t
. Hence, there must be a
ʔ
∈
D
sub
(
sCSP
)
such that
ʔ
a
−ₒ
ʔ
and [
T
|
Act
ʔ
]
d
(
T
,
ʔ
).
⃒
⃒
ʓ
. It follows that
o
∈
A
d
(
T
i
,
ʔ
) for
i
5. (
⃐
) Assume
o
i
∈
A
=
1, 2. Then, [
T
i
|
Act
ʔ
]
⃒
ʓ
i
for some stable
ʓ
i
with $
ʓ
i
=
o
i
. By Theorem
6.5
(i) we have
[(
T
1
p
↕
T
2
)
|
Act
ʔ
]
=
p
·
[
T
1
|
Act
ʔ
]
+
(1
−
p
)
·
[
T
2
|
Act
ʔ
]
⃒
p
·
ʓ
1
+
(1
−
p
)
·
ʓ
2
,
and
p
·
ʓ
1
+
(1
−
p
)
·
ʓ
2
is stable. Moreover,
$(
p
·
ʓ
1
+
(1
−
p
)
·
ʓ
2
)
=
p
·
o
1
+
(1
−
p
)
·
o
2
,
∈
A
d
(
T
1
p
↕
so
o
T
2
,
ʔ
).
⃒
∈
A
d
(
T
1
p
↕
=
(
) Suppose
o
T
2
,
ʔ
). Then, there is a stable
ʓ
with $
ʓ
o
such that [(
T
1
p
↕
T
2
)
|
Act
ʔ
]
=
p
·
[
T
1
|
Act
ʔ
]
+
(1
−
p
)
·
[
T
2
|
Act
ʔ
]
⃒
ʓ
.By
Theorem
6.5
(ii), there are
ʓ
i
for
i
=
1, 2, such that [
T
i
|
Act
ʔ
]
⃒
ʓ
i
and also
d
(
T
i
,
ʔ
) for
ʓ
=
p
·
ʓ
1
+
(1
−
p
)
·
ʓ
2
.As
ʓ
1
and
ʓ
2
are stable, we have $
ʓ
i
∈
A
i
=
1, 2. Moreover,
o
=
$
ʓ
=
p
·
$
ʓ
1
+
(1
−
p
)
·
$
ʓ
2
.
6. Suppose
q
∈
[0, 1] and
ʔ
1
,
ʔ
2
∈
D
sub
(
rpCSP
) with
ʔ
⃒
q
·
ʔ
1
+
(1
−
q
)
·
ʔ
2
d
(
T
i
,
ʔ
i
). Then there are stable
ʓ
i
with [
T
i
|
Act
ʔ
i
]
and
o
i
∈
A
⃒
ʓ
i
and $
ʓ
i
=
o
i
.Now
[(
T
1
T
2
)
|
Act
ʔ
]
⃒
q
·
[(
T
1
T
2
)
|
Act
ʔ
1
]
+
(1
−
q
)
·
[(
T
1
T
2
)
|
Act
ʔ
2
]
˄
−ₒ
q
·
[
T
1
|
Act
ʔ
1
]
+
(1
−
q
)
·
[
T
2
|
Act
ʔ
2
]
⃒
q
·
ʓ
1
+
(1
−
q
)
·
ʓ
2
The latter subdistribution is stable and satisfies
$(
q
·
ʓ
1
+
(1
−
q
)
·
ʓ
2
)
=
q
·
o
1
+
(1
−
q
)
·
o
2
.
d
(
T
1
Hence
q
·
o
1
+
(1
−
q
)
·
o
2
∈
A
T
2
,
ʔ
).
We also have the converse to part (6) of this lemma by mimicking Lemma 5.10.
For that purpose, we use two technical lemmas whose proofs are similar to those for
Lemmas
6.28
and
6.29
, respectively.
˄
−ₒ
Lemma 6.40
Suppose ʔ
|
A
(
T
1
T
2
)
ʓ . Then there exist subdistributions
ʔ
ₒ
,
ʔ
1
,
ʔ
2
,
ʔ
next
(possibly empty) such that
ʔ
1
+
ʔ
2
ʔ
ₒ
+
(i) ʔ
=
˄
−ₒ
(ii) ʔ
ₒ
ʔ
next
ʔ
1
|
A
T
1
+
ʔ
2
|
A
T
2
ʔ
next
(iii) ʓ
=
|
A
(
T
1
T
2
)
+
˄
−ₒ
Proof
By Lemma
6.1
ʔ
|
A
(
T
1
T
2
)
ʓ
implies that
˄
−ₒ
ʔ
=
p
i
·
s
i
,
s
i
|
A
(
T
1
T
2
)
ʓ
i
,
ʓ
=
p
i
·
ʓ
i
,
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