\section{Theorems and Proofs}\label{sec:Theorems}
As we shall see in this section, everything in the previous section holds
true for braids, and hence we get a nice OU theorem for braids, ``every
braid, when considered as a tangle, has a unique reduced OU form'', so
braids can be be separated by their OU forms. As we shall also see here,
everything in the previous section extends to virtual tangles, and is true
when restricted to virtual braids. Thus we find that every virtual braid
has a unique OU form when it is regarded as a virtual tangle; with extra
work we find that this OU form is a complete invariant of virtual braids.
\draftcut \input c.tex
\draftcut \input v.tex
\draftcut \input a.tex