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Proof.
In [5], we proved that for any
x, y, z
∈
V
(PS
3
)
,
x
=
y
∧
y
=
z
≤
x
=
z
.
Let
x
and
y
be two
ʱ
-like elements in
V
(PS
3
)
.So
x
=
ʱ
∧
ʱ
=
y
≤
x
=
y
.
By Lemma 7 we have
x
=
ʱ
=1=
y
=
ʱ
, which implies
x
=
y
=1.
Conversely let
x
=
y
= 1. By similar argument we can write,
x
=
y
∧
x
=
ʱ
≤
y
=
ʱ
and hence
y
=
ʱ
= 1. Again by Lemma 7 it can be concluded
that
y
is
ʱ
-like.
V
(PS
3
)
be ʱ-like for some non-zero ʱ
Theorem 10.
Let x
∈
∈
ORD
. For any
V
(PS
3
)
,
y
∈
y
∈
x
∈{
1
,
1
/
2
}
if and only if y is ʲ-like for some ʲ
∈
ʱ.
Proof.
Let
y
be
ʲ
-like for some
ʲ
∈
ʱ
.Now
=
u∈
dom(
x
)
y
∈
x
(
x
(
u
)
∧
u
=
y
)
≥
x
(
v
)
∧
v
=
y
,
where
v
∈
dom(
x
)is
ʲ
-like and
x
(
v
)
∈{
1
,
1
/
2
}
≥
1
/
2
,
by Theorem 9.
Conversely, let
y
∈
x
∈{
1
,
1
/
2
}
, i.e.,
(
x
(
u
)
∧
u
=
y
)
∈{
1
,
1
/
2
}
.
u∈
dom(
x
)
Hence there exists some
ʲ
-like
v
∈
dom(
x
) such that
x
(
v
)
∈{
1
,
1
/
2
}
and
v
=
y
=1,where
ʲ
∈
ʱ
. So by Theorem 9, it follows that
y
is also
ʲ
-like.
be defined on
V
(PS
3
)
by
x
Let a binary class relation
∼
∼
y
if and only
if
V
(PS
3
)
|
=
x
=
y
, i.e.,
x
=
y
= 1. This relation is discussed in [5] where
∼
is an
equivalence class relation
. Theorem 9 shows for
it is mentioned that
each
ʱ
∈
ORD the collection of all
ʱ
-like elements forms an equivalence class in
V
(PS
3
)
/
.If
x
and
y
are two elements in the classes of
ʱ
-like and
ʲ
-like elements
in
V
(PS
3
)
/
∼
∼
then
ʱ
∈
ʲ
is true in
V
implies
x
∈
y
is valid in
V
(PS
3
)
.
4Ordin sinV
(PS
3
)
We now rewrite the definitions of
§
2 in the language of set theory:
Trans(
x
)=
∀y∀z
(
z ∈ y ∧ y ∈ x → z ∈ x
)
y
))
4
LO(
x
)=
∀
y
∀
z
((
y
∈
x
∧
z
∈
x
)
→
(
y
∈
z
∨
y
=
z
∨
z
∈
WO
∈
(
x
)=LO(
x
)
∧∀
y
(
y
ↆ
x
∧¬
(
y
=
∅
)
→∃
z
(
z
∈
y
∧
z
∩
y
=
∅
))
ORD(
x
)=Trans(
x
)
∧
WO
∈
(
x
)
4
LO(
x
) stands for the formula:
x
is a linear orderdered set with respect to
∈
.
