Let $P_n := \{ p \in \mathbb{R}^n : p_i > 0 \wedge p_1+...+p_n =1 \}$. Let $A \in \mathbb{R}^{m \times n}$.
S1: $\exists \xi \in \mathbb{R}^{1 \times m} : \xi A > 0 \Leftrightarrow \neg \exists p \in P_n : Ap =0$.
Hahn-Banach: Every nonzero bounded linear functional u on a subspace M of a separable normed linear space E, whose kernel is located in E, has an extension v with $\| u \| = \|v\|$.
Is there a constructive proof that S1 implies Hahn Banach?