  We give a complete proof of a propagation theorem of multiplicity-free
property from fibers to spaces of global sections for holomorphic vector
bundles. The propagation theorem is formalised in three ways, aiming for
producing various multiplicity-free theorems in representation theory for both
finite and infinite dimensional cases in a systematic and synthetic manner.
  The key geometric condition in our theorem is an orbit-preserving
anti-holomorphic diffeomorphism on the base space, which brings us to the
concept of visible actions on complex manifolds.
