  The generalized gluing and resmoothing theorem originally proved by LeClair,
Peskin and Preitschopf, gives a powerful formula for the fused vertex obtained
by contracting any two vertices in string field theories. Although the theorem
is naturally expected to hold for the vertices at any loop level, the original
proof was restricted to the vertices at tree level. Here we present a
simplified proof for the tree level theorem and then prove explicitly the
extended version at one-loop level. We also find that a non-trivial sign
factor, which depends on the string states to be contracted, appears in the
theorem. This sign factor turns out to be essential for reproducing correctly
the conformal field theory correlation function on the torus.
