Lemma 7.10.
Assuming that:
X
1
,
X
2
,
Y
1
,
Y
2
satisfy:
X
1
and
X
2
are copies of
X
,
Y
1
and
Y
2
are copies of
Y
, and all of them are independent
Then
6
d
[
X
;
Y
]
≥
2
d
[
X
1
+
X
2
;
Y
1
+
Y
2
]
+
d
[
X
1
+
Y
2
;
X
2
+
Y
1
]
+
2
d
[
X
1
|
X
1
+
X
2
;
Y
1
|
Y
1
+
Y
2
]
+
d
[
X
1
|
X
1
+
Y
1
;
X
2
|
X
2
+
Y
2
]
+
2
3
d
[
X
1
+
X
2
;
X
1
+
Y
1
∥
X
2
+
Y
1
,
X
1
+
Y
2
]
+
1
3
d
[
X
1
+
Y
1
;
X
1
+
Y
2
∥
X
1
+
X
2
,
Y
1
+
Y
2
]
OR? TODO: figure out which is correct
6
d
[
X
;
Y
]
≥
2
d
[
X
1
+
X
2
;
Y
1
+
Y
2
]
+
d
[
X
1
+
Y
1
;
X
2
+
Y
2
]
+
2
d
[
X
1
|
X
1
+
X
2
;
Y
1
|
Y
1
+
Y
2
]
+
d
[
X
1
|
X
1
+
Y
1
;
X
2
|
X
2
+
Y
2
]
+
2
3
d
[
X
1
+
X
2
|
X
1
+
Y
1
;
X
2
+
Y
1
|
X
1
+
Y
2
]
+
1
3
d
[
X
1
+
Y
1
|
X
1
+
Y
2
;
X
−
1
+
X
1
|
Y
1
+
Y
2
]