by contradiction, suppose that $-tx_0\in \Omega$ for some $t>0$. Then it follows from [@BG Lemma 3.1] that $$0=\frac{t}{1+t}x_0+\frac{1}{1+t}\big(-tx_0\big)\in\mbox{\rm iri}(\Omega),$$ which clearly contradicts the imposed assumption on $0\notin{\rm iri}(\Omega)$. Denoting now $x_k:=-(x_0/k)\in -\Omega$ tells us that $x_k\notin{\Omega}$ for every $k\in\N$ and that $x_k\to 0$ as $k\to\infty$.$\h$

The second lemma, which is broadly employed in the subsequent material, provides an equivalent description of quasi-relative interiors via [*proper separation*]{} of convex sets in the general LCTV space framework.

[**(proper separation description of quasi-relative interiors in LCTV spaces).**]{}\[proper separation\] Let $\Omega$ be a convex set in an LCTV space $X$ with $\ox\in\Omega$. Then $\ox\notin\mbox{\rm qri}(\Omega)$ if and only if the sets $\{\ox\}$ and $\Omega$ can be properly separated.

[**Proof.**]{} As stated in Theorem \[Thrpriqri\], $\ox\in\mbox{\rm qri}(\Omega)$ if and only if the normal cone $N(\ox;\Omega)$ is a subspace of $X^*$. Thus we get that $\ox\notin\mbox{\rm qri}(\Omega)$ if and only if there exists $x^*\in N(\ox;\Omega)$ with $-x^*\notin N(\ox;\Omega)$. It follows from the normal cone definition that $\la x^*,x\ra\le\la x^*,\ox\ra$ for all $x\in\Omega$. Then the inclusion $-x^*\notin N(\ox;\Omega)$ gives us such $x_0\in\Omega$ that $\la-x^*,x_0\ra>\la-x^*,\ox\ra$, which reads as $\la x^*,x_0\ra<\la x^*,\ox\ra$ and hence justifies the claim. $\h$

The third lemma provides a useful version of the [*strict separation*]{} 