t
y
1
b
N
˜ C
N −1
b
N +1
e t
y
1
b
N
˜ C
N −1
b
N +1
e
Figure 12: Schematic representations of the events a E
′
b
N
, b
N +1
; ˜ C
N −1
∩ G
1
t
y 1
b
N
, e; ˜ C
N −1
and b E
′
b
N
, b
N +1
; ˜ C
N −1
∩ G
2
t
y 1
b
N
, e; ˜ C
N −1
. of
E
t
y 1
b
N
, e; ˜ C
N −1
, which is an increasing event. Therefore, similarly to the analysis in 6.48, we use Lemma 6.6 to obtain
X
b
N
,b
N +1
,e b
N +1
6=e
p
b
N
p
b
N +1
p
e
M
N
b
N
˜ E
b
N +1
h
1
E
′
b
N
,b
N +1
;˜ C
N −1
1
G
2 t y1
b
N
,e;˜ C
N −1
B
N1
δ
b
N +1
, y
1
; C
N
i δ
e,y
2
≤ X
b
N
,b
N +1
,e b
N +1
6=e
p
b
N
p
b
N +1
p
e
M
N
b
N
E h
1
E
′
b
N
,b
N +1
;˜ C
N −1
1
E
t y1
b
N
,e;˜ C
N −1
B
N1
δ
b
N +1
, y
1
; C
N
i δ
e,y
2
= X
b
N +1
,e b
N +1
6=e
p
b
N +1
p
e
M
N +1
b
N +1
1
E
t y1
b
N
,e;˜ C
N −1
B
N1
δ
b
N +1
, y
1
; C
N
δ
e,y
2
. 6.52
The bound 6.18 for N
2
= 0 now follows from Lemma 6.7. This completes the proof of Lemma 6.3 for N
2
= 0. Proof of Lemma 6.3 for N
2
≥ 1. First we prove the bound on φ
N ,N1,1
y
1
, y
2
+
, where, by 6.39– 6.40, 6.3 and 5.40,
φ
N ,N1,1
y
1
, y
2
+
≤ X
b
N +1
,e b
N +1
6=e
p
b
N +1
p
e
˜ M
N +1
b
N +1
1
V
t y1 −ǫ
b
N
,e
B
N1
δ
b
N +1
, y
1
; C
N
B
δ
e, y
2
; ˜ C
e
N
. 6.53
Following the argument around 5.47–5.49 , we have 6.53
≤ X
b
N +1
,e,e
′
e
′
=y
2
X
c
p
b
N +1
M
N +1
b
N +1
1
V
t y1 −ǫ
b
N
,e
∩ {c∈˜C
e N
}
B
N1
δ
b
N +1
, y
1
; ˜ C
N
p
e
P e, e
′
; c p
e
′
, 6.54
865
where ˜ C
N
= ˜ C
b
N +1
b
N
. By 6.45 with ~x = c and 5.38, we obtain
6.54 ≤
X
b= · ,y
1
e
′
= · ,y
2
p
b
p
e
′
X
η
X
c
X
e
R
N +N1
b, e; ℓ
η
c p
e
P e, e
′
; c
| {z
}
R
N +N1
b ,y;2
1
y
c,2
c
e
′
= X
b= · ,y
1
e
′
= · ,y
2
p
b
p
e
′
R
N +N1
b, y; E
y
e
′
.
6.55 This shows that
φ
N ,N1,1
y
1
, y
2
+
≤ X
u
1
,u
2
p
ǫ
y
1
− u
1
p
ǫ
y
2
− u
2
R
N +N1,1
u
1
, u
2
, 6.56
as required. To extend the proof of 6.17 to all N
2
, we estimate B
N2
δ
e, y
2
; ˜ C
e
N
using 5.40. Since the bound on B
N2
δ
e, y
2
; ˜ C
e
N
is the same as N
2
− 1 applications of Construction E to P e, u
2
; ˜ C
e
N
, the bound follows by the definition of R
N +N1,N2
y
1
, y
2
. The proof of 6.18 for
φ
N ,N1,N2
y
1
, y
2
−
proceeds similarly, when we use 6.46 rather than 6.45. This completes the proof of Lemma 6.3.
6.3.2 Proof of Lemma 6.7
Proof of Lemma 6.7 for N
1
= 0. Since B
δ
b
N +1
, y
1
; ˜ C
N
= δ
b
N +1
,y
1
and therefore V
t
y 1
−ǫ
and E
t
y 1
can be replaced by V
t
bN+1
and E
t
bN+1
+ ǫ
, respectively, LHS of 6.45 =
X
b
N +1
= · ,y
1
p
b
N +1
M
N +1
b
N +1
1
V
t b N +1
b
N
,y
2
∩ {~x ∈˜C
N
}
, LHS of 6.46 =
X
b
N +1
= · ,y
1
p
b
N +1
M
N +1
b
N +1
1
E
t b N +1
+ ǫ
b
N
,y
2
;˜ C
N −1
∩ {~x ∈˜C
N
}
. Recalling the definitions of R
N
and Q
N
in 6.13–6.14, we can prove Lemma 6.7 for N
1
= 0 by showing
M
N +1
b
N +1
1
V
t b N +1
b
N
,y
2
∩ {~x ∈˜C
N
}
≤ P
N
b
N +1
; V
t
bN+1
y
2
, ℓ~x
N ≥ 0, 6.57
M
N +1
b
N +1
1
E
t b N +1
+ ǫ
b
N
,y
2
;˜ C
N −1
∩ {~x ∈˜C
N
}
≤ P
N
b
N +1
; E
t
bN+1
y
2
, ℓ~x
N ≥ 1. 6.58
By the nested structure of M
N +1
b
N +1
cf., 3.27, LHS of 6.57 =
X
b
N
p
b
N
M
N
b
N
M
1
b
N
,b
N +1
;˜ C
N −1
1
V
t b N +1
b
N
,y
2
∩ {~x ∈˜C
N
}
, 6.59
LHS of 6.58 = X
b
N −1
p
b
N −1
M
N −1
b
N −1
M
2
b
N −1
,b
N +1
;˜ C
N −2
1
E
t b N +1
+ ǫ
b
N
,y
2
;˜ C
N −1
∩ {~x ∈˜C
N
}
. 6.60
866
On the other hand, by the recursive definition of P
N
cf., 5.58, RHS of 6.57 =
X
b
N
p
b
N
X
c
P
N −1
b
N
;
ℓc P
b
N
, b
N +1
; c, V
t
bN+1
y
2
,
ℓ~x
, 6.61
RHS of 6.58 = X
b
N −1
p
b
N −1
X
c
P
N −2
b
N −1
;
ℓc P
1
b
N −1
, b
N +1
; c,
E
t
bN+1
y
2
, ℓ~x
, 6.62
where Construction ℓc in 6.61 is applied to the N −1
th
admissible lines of P
N
and that in 6.62 is applied to the N
− 2
th
admissible lines. By comparing the above expressions and following the argument around 5.47–5.49, it thus suffices to prove
M
1
b
N
,b
N +1
;˜ C
N −1
1
V
t b N +1
b
N
,y
2
∩ {~x ∈˜C
N
}
≤ P b
N
, b
N +1
; ˜ C
N −1
, V
t
bN+1
y
2
, ℓ~x
, 6.63
M
2
b
N −1
,b
N +1
;˜ C
N −2
1
E
t b N +1
+ ǫ
b
N
,y
2
;˜ C
N −1
∩ {~x ∈˜C
N
}
≤ P
1
b
N −1
, b
N +1
; ˜ C
N −2
, E
t
bN+1
y
2
, ℓ~x
. 6.64
First we prove 6.63. Note that, by 3.16, LHS of 6.63 = P
E
′
b
N
, b
N +1
; ˜ C
N −1
∩ V
t
bN+1
b
N
, y
2
∩ {~x ∈ ˜C
N
} .
6.65 Using 6.40, we obtain
V
t
bN+1
b
N
, y
2
⊆ [
v
t
v
=t
bN+1
[
z
n {b
N
−→ z} ◦ {z −→ v} ◦ {v −→ y
2
} ◦ {z −→ y
2
} o
, 6.66
hence 6.65
≤ X
v
t
v
=t
bN+1
P E
′
b
N
, b
N +1
; ˜ C
N −1
∩ {~x ∈ ˜C
N
}
∩ [
z
n {b
N
−→ z} ◦ {z −→ v} ◦ {v −→ y
2
} ◦ {z −→ y
2
} o
. 6.67
The event E
′
b
N
, b
N +1
; ˜ C
N −1
implies that there are disjoint connections necessary to obtain the bounding diagram P
b
N
, b
N +1
; ˜ C
N −1
. The event {b
N
−→ v} =
S
z
{{b
N
−→ z} ◦ {z −→ v}}
can be accounted for by an application of Construction ℓv, and then {v −→ y
2
} ◦ {z −→ y
2
} can be accounted for by an application of Construction 2
v
y
2
. The event {~x ∈ ˜C
N
} implies additional connections, accounted for by an application of Construction
ℓ~x . By 6.13, this completes the
proof of 6.63. Next, we prove 6.64. Note that, by 3.19,
LHS of 6.64 = X
b
N
p
b
N
M
1
b
N −1
,b
N
;˜ C
N −2
P E
′
b
N
, b
N +1
; ˜ C
N −1
∩ E
t
bN+1
+ ǫ
b
N
, y
2
; ˜ C
N −1
∩ {~x ∈ ˜C
N
} .
6.68
867
Using 6.43 and following the argument below 6.67, we obtain P
E
′
b
N
, b
N +1
; ˜ C
N −1
∩ E
t
bN+1
+ ǫ
b
N
, y
2
; ˜ C
N −1
∩ {~x ∈ ˜C
N
} ≤ P
E
′
b
N
, b
N +1
; ˜ C
N −1
∩ [
c ,w
∈˜C
N −1
[
z
∈Λ t
z
t
bN+1
n {b
N
−→ z} ◦ {z −→ w } ◦ {w −→ y
2
} ◦ {z −→ y
2
} o
∩ n
{c = w , z 6−→ w
−
} ∪ {c 6= w
−
, c, w ∈ ˜C
N −1
} o
∩ {~x ∈ ˜C
N
}
. 6.69
Similarly to the above, E
′
b
N
, b
N +1
; ˜ C
N −1
implies the existence of disjoint connections necessary to obtain the bounding diagram P
b
N
, b
N +1
; ˜ C
N −1
. The event subject to the union over z is accounted for by an application of Construction Bu followed by multiplication of the sum of
S
u, w ; ˜ C
