Lemma 7.7
(Fibring lemma)
.
Assuming that:
G
and
H
are abelian groups
ϕ
:
G
→
H
a homomorphism
let
X
,
Y
independent
G
-valued random variables
Then
d
[
X
;
Y
]
=
d
[
ϕ
(
X
)
;
ϕ
(
Y
)
]
+
d
[
X
|
ϕ
(
X
)
;
Y
|
ϕ
(
Y
)
]
+
I
[
X
−
Y
:
ϕ
(
X
)
,
ϕ
(
Y
)
|
ϕ
(
X
)
−
ϕ
(
Y
)
]
.