A corollary to the Reeh-Schlieder theorem is proved: that the time-ordered Vacuum Expectation Values and the S-matrix of a regularized Lagrangian quantum theory can be approximated by a local operator that uses nonlinear functionals of a locally supported source function. The use of operator-valued nonlinear functionals of the source effectively introduces many abstract fragments of the source to give a top-down construction of interacting quantum fields, in contrast to a bottom-up blocking and scaling construction.