= Coordinate-splitting proof of the BK inequality
For <increasing events> $A,B$ under a finite <Bernoulli distribution> <product measure>, first assume all parameters are at most $1/2$. Write $A_0\subseteq A_1$ and $B_0\subseteq B_1$ for last-coordinate sections. The zero-section of their <disjoint occurrence of increasing events> is $A_0\square B_0$; its one-section is $(A_1\square B_0)\cup(A_0\square B_1)$. The intersection in this union contains $A_0\square B_0$. If $a_i=\mathbb P(A_i)$, $b_i=\mathbb P(B_i)$ and the last parameter is $p\leq1/2$, <mathematical induction> bounds disjoint occurrence by $(1-2p)a_0b_0+p a_1b_0+p a_0b_1$. The product of the two marginal <probabilities> exceeds this bound by $p^2(a_1-a_0)(b_1-b_0)\geq0$. For a larger parameter $p_i<1$, replace that coordinate by the <logical disjunction> of $m_i$ <independent> bits with parameter $1-(1-p_i)^{1/m_i}\leq1/2$. Disjoint original witnesses lift to disjoint bit witnesses. Apply the small-parameter inequality and then pass to degenerate parameters by continuity.
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