By Lemmas 35 and 36, \(P\) is a \(\sigma \)–closed convex cone in the real locally convex space \(M^{s}\). By the Hahn-Banach Separation Theorem, there is a \(\sigma \)–continuous real linear functional \(g\) on \(M^{s}\) such that \(\inf _{h\in P}g(h){\gt}g(a)\). Since \(P\) is a cone, if \(g(h){\lt}0\) then we could scale \(h\) by a positive constant so \(g(ch)\le g(a)\), which is nonsense. Therefore \(g(h)\ge 0\) for all \(h\ge 0\), and the infimum above must be zero (which can again by seen by scaling). It follows that \(0{\gt}g(a)\). To appropriately extend \(g\) to a functional on \(M\), define \(\varphi (a + ib)=g(a)+ig(b)\) for any \(a,b \in M^{s}\). This \(\varphi \) is a (complex) linear functional on \(M\), and the \(*\)–operation is \(\sigma \)–continuous because \(M^{s}\) is \(\sigma \)–closed (by Lemma 36). It follows that \(\varphi \) is a \(\sigma \)–continuous positive linear functional on \(M\) such that \(\varphi (a)=g(a){\lt}0\).

Given nonzero \(a\in P\) in \(M\), since \(P\) is a cone, \(-a \notin P\). By Lemma 37, there is a \(\varphi \in T\) such that \(\varphi (-a){\lt}0\), hence \(\varphi (a){\gt}0\). The desired statement follows by contraposition.