direct consequence of Theorem \[Lemalinearmap\].

[**(quasi-relative interiors of set differences).**]{}\[Corrosub\] Let $\Omega_1$ and $\Omega_2$ be convex subsets of an LCTV space $X$. Suppose that $\iri(\Omega_1)\ne\emp$, $\iri(\Omega_2)\ne\emp$, and $\Omega_1-\Omega_2$ is quasi-regular. Then we have the equality $$\qri(\Omega_1-\Omega_2)=\qri(\Omega_1)-\qri(\Omega_2).$$

[**Proof.**]{} Define the linear continuous mapping $A\colon X\times X\to X$ by $A(x,y):=x-y$, and let $\Omega:=\Omega_1\times\Omega_2$. Then $\mbox{\rm iri}(\Omega)=\mbox{\rm iri}(\Omega_1)\times\mbox{\rm iri}(\Omega_2)\ne\emp$; see [@bl Proposition 2.5]. Applying Theorem \[Lemalinearmap\] gives us $$\mbox{\rm qri}(\Omega_1-\Omega_2)=A\big(\qri(\Omega)\big)=\qri\big(A(\Omega)\big)=\qri(\Omega_1)-\mbox{\rm qri}(\Omega_2),$$ and thus we arrive at the claimed equality. $\h$

[**(proper separation in LCTV spaces via quasi-relative interiors).**]{}\[Theoqricap\] Let $\Omega_1$ and $\Omega_2$ be convex subsets of an LCTV space $X$. Assume that $\iri(\Omega_1)\ne\emp$, $\iri(\Omega_2)\ne\emp$, and $\Omega_1-\Omega_2$ is quasi-regular. Then $\Omega_1$ and $\Omega_2$ are properly separated if and only if $$\label{qri-sep}
\qri(\Omega_1)\cap\qri(\Omega_2)=\emp.$$

[**Proof.**]{} Define $\Omega:=\Omega_1-\Omega_2$ and get from Corollary \[Corrosub\] that condition reduces to $$0\notin\qri(\Omega_1-\Omega_2)=\qri(\Omega_1)-\qri(\Omega_2).$$ If holds, then $0\notin\qri(\Omega_1-\Omega_2)=\qri(\Omega)$. Lemma \[proper separation\] tells us that the sets $\Omega$ and $\{0\}$ are properly separated, which clearly ensures the proper separation of the sets $\Omega_1$ and $\Omega_2$.

To verify the opposite implication, suppose that $\Omega_1$ and $\Omega_2$ are properly separated, which implies that the sets $\Omega=\Omega_1-\Omega_2$ and $\{0\}$ are properly separated as well. Then using Lemma \[proper separation\] and Corollary \[Corrosub\] tells us that $$0\notin\qri(\Omega)=\qri(\Omega_1-\Omega_2)=\qri(\Omega_1)-\qri(\Omega_2),$$ and 