= Addition of filters on the natural numbers
{title2=$A\in\mathcal F+\mathcal G\Longleftrightarrow\forall_{\mathcal F}x\,\forall_{\mathcal G}y\,(x+y\in A)$}
For proper filters on positive integers, define $A\in\mathcal F+\mathcal G$ when $\{x:A-x\in\mathcal G\}\in\mathcal F$. This is again a proper filter. The inner truth-set map preserves the whole set, the empty set, inclusions and finite intersections; applying the outer filter gives all filter axioms. The order of the two filter quantifiers is part of the definition and is not silently interchangeable.
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