  We generalize a fundamental theorem in higher dimensional value distribution
theory about entire curves in subvarieties $X$ of semi-abelian varieties to the
situation of the sequences of holomorphic maps from the unit disc into $X$.
This generalization implies, among other things, that subvarieties of log
general type in semi-abelian varieties are pseudo-Kobayashi hyperbolic. As
another application, we improve a classical theorem due to Cartan in 1920's
about the system of nowhere vanishing holomorphic functions on the unit disc
satisfying Borel's identity.
