On Families of Complex Lines Sufficient for Holomorphic Extension

It is shown that the set $\mathfrak L_\Gamma$ of all complex lines passing through a germ of a generating manifold $\Gamma$ is sufficient for any continuous function $f$ defined on the boundary of a bounded domain $D\subset\mathbb C^n$ with connected smooth boundary and having the holomorphic one-dimensional extension property along all lines from $\mathfrak L_\Gamma$ to admit a holomorphic extension to $D$ as a function of many complex variables.