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Write $a=P(A_1)=P(A_2)$. The complements are decreasing events. The Harris-FKG inequality for a product measure givesThe same correlation inequality for two decreasing events follows from that for their increasing complements, since their covariance is unchanged. ThusBoth $1-a$ and the square root are nonnegative, so rearranging gives the square-root bound for increasing events:No independence between the events themselves is asserted; independence is the coordinate property of the underlying product measure.

Let $E_T,E_B,E_L,E_R$ be the increasing events that an infinite simple open ray in a graph starts at the indicated side and immediately leaves the box, with all later graph vertices outside it. A corner exit is assigned according to the direction of its outgoing edge. The four events have the same probability $a_N$ by rotational symmetry. Their union is $D_N$ from the preceding step.

The Harris-FKG inequality applies to these exterior-ray events by approximation with finite connections to larger boxes and passage to decreasing limits. Positive association of the four decreasing events $E_s^c$, by iterating the Harris-FKG inequality, givesEquivalently, one may apply the preceding square-root bound for increasing events first to the two opposite-side unions and then to the two individual sides of one pair. ThereforeA union bound now showsChoose $N$ such that $1-\mathbb{P}(D_N)<(1/8{)}^4$. ThenThe infinite exterior clusters touched by these two sides need not be the same. The same argument works for dual boxes and any sufficiently large size; this permits the matching primal/dual contours used in the full proof.