= Section formula for a grid neighbourhood
Let $A_j\subseteq[k]^{d-1}$ be the section of $A\subseteq[k]^d$ with one coordinate fixed at $j$, and put $A_0=A_{k+1}=\varnothing$. The corresponding section of its <closed graph neighbourhood> is
$$
N[A]_j=N[A_j]\cup A_{j-1}\cup A_{j+1}.
$$
After coordinate compression, the three sets on the right are nested initial simplicial segments. Their union therefore has the largest of their three cardinalities, which is no larger than the original union.
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