  We consider a problem of whether a property of holomorphic curves on a subset
$X$ of the complex plane can be extended to the whole complex plane. In this
paper, the property we consider is uniqueness of holomorphic curves. We
introduce the propagation set. Simply speaking, $X$ is a propagation set if
linear relation of holomorphic curves on the part of preimage of hyperplanes
contained in $X$ can be extended to the whole complex plane. If the holomorphic
curves are of infinite order, we prove the existence of a propagation set which
is the union of a sequence of disks (In fact, the method applies to the case of
finite order). For a general case, the union of a sequence of annuli will be a
propagation set. The classic five-value theorem and four-value theorem of R.
Nevanlinna are established in such propagation sets.
