On embedding some $G$-filtered rings into skew fields

We consider the filtered rings with filtration $v$ taking values in an ordered group $G$ (or $G$-filtered rings). We prove that if a ring $R$ of this type satisfies the condition
$$ \forall a,b\in R^*\quad\forall\varepsilon\in G\quad\exists x,y\in R^*\qquad v(a\cdot x-b\cdot y)>\varepsilon\cdot v(a\cdot x) $$
then $R$ embeds into a skew field. This skew field $D$ becomes a topological ring in the topology induced by an extension of $v$, while $R\cdot R^{-1}$ is everywhere dense in $D$.