  Based on an analogue for systems of partial isomorphisms between lower
sections in a complemented modular lattice we prove that principal right ideals
$aR \cong bR$ in a (von Neumann) regular ring $R$ are perspective if
  $aR \cap bR$ is of finite height in $L(R)$. This is applied to derive, for
existence-varieties $\mathcal{V}$ of regular rings, equivalence of
unit-regularity and direct finiteness, both conceived as a property shared by
all members of $\mathcal{V}$.
