= One-dimensional dominated extension of a real linear functional
{title2=$\sup_m(g(m)-p(m-v))\leq c\leq\inf_m(p(m+v)-g(m))$}
A real <linear functional> $g$ on a <vector subspace>, dominated by a <sublinear function> $p$, can be extended across one new vector $v$ by setting $\widetilde g(m+tv)=g(m)+tc$. The displayed lower and upper bounds are compatible because $g(m+n)\leq p(m-v)+p(n+v)$. A value between them preserves domination for both signs of $t$. Chain unions and <Zorn's lemma> then yield the full <Hahn-Banach theorem>.
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