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\centerline{Additions to Part I: Numerical List of Forms}
\smallskip
\item{}{\bf [43 AI]} Every countably well-founded relation is
well-founded. (A relation $R$ is {\it (countably) well-founded}
if every (countable) set in the domain of $R$ has an $R$-minimal
element.) \ac{Diener} \cite{1994}.
\smallskip
Add a reference to Keremedis in form [1 DG].\newline
\item{}{\bf [1 DG]} Vector Space Kinna-Wagner Principle: For
every family $V = \{V_i : i \in K\}$ of non-trivial vector spaces
there is a family $F = \{F_i : i\in K\}$ such that for each
$i\in K$, $F_i$ is a non-empty, independent subset of $V_i$.
\ac{Keremedis} \cite{1999e} and note 127.
\medskip
\item{}{\bf [14 DA]} Keimel's Representation Theorem. Any hyperarchimedian
$l$-group can be imbedded into a Boolean product of a family of simple,
abelean $l$-groups. \ac{Gluschankof} \cite{1995} and note 151.
\iput{hyperarchimedian $l$-group}
\medskip
\item{}{\bf [14 DB]} If $G$ is an \icopy{achimedian $l$-group} with a weak
unit, then $G$ can be imbedded into $D(X)$ where $X$ is a locally compact
topological space and $D(X)$ is the set of continuous maps
$f: X\to \Bbb R \cup \{-\infty, \infty\}$ such that the inverse image
of $\Bbb R$ is dense in $X$. \ac{Gluschankof} \cite{1995} and note 151.
\medskip
\item{}{\bf [14 DC]} \icopy{Clifford's Theorem}. An \icopy{abelean
$l$-group} is isomorphic to a subdirect product of totally ordered
abelian groups. \ac{Gluschankof} \cite{1995} and note 151.
\medskip
\item{}{\bf [14 DD]} \icopy{Lorenzen's Theorem}. A representable
$l$-group is isomorphic to a subdirect product of totally ordered
groups. \ac{Gluschankof} \cite{1995} and note 151.
\medskip
\item{}{\bf [14 DE]} \icopy{Holland's Theorem}. Any $l$-group is
isomorphic to a subdirect product of transitive $l$-groups.
\ac{Gluschankof} \cite{1995} and note 151.
\medskip
\item{}{\bf [14 DF]} Any $l$-group can be imbedded into the $l$-preserving
permutations of a totally ordered set. \ac{Gluschankof} \cite{1995} and
note 151.
\medskip
\item{}{\bf [14 DG]} In any hyperarchimedian $l$-group any proper
$l$-ideal can be extended to a prime one. \ac{Gluschankof} \cite{1989}
and note 151.
\medskip
\noindent{\bf FORM 408.} If $\{f_i: i\in I\}$ is a family of functions
such that for each $i\in I$, $f_i\subseteq E\times W$, where $E$ and
$W$ are non-empty sets, and $\Cal B$ is a filter base on $I$ such that
\item {1.} For all $B\in\Cal B$ and all finite $F\subseteq E$ there
is an $i\in I$ such that $f_i$ is defined on $F$, and
\item {2.} For all $B \in\Cal B$ and all finite $F\subseteq E$ there
exist at most finitely many functions on $F$ which are restrictions
of the functions $f_i$ with $i\in I$,
\noindent
then there is a function $f$ with domain $E$ such that for each
finite $F\subseteq E$ and each $B\in\Cal B$ there is an $i\in I$
such that $f|F = f_i|F$. \ac{Felscher} \cite{1964}.
\medskip
\noindent{\bf FORM 409.} Suppose $(G,\Gamma)$ is a locally finite
graph (i.e. $G$ is a non-empty set and $\Gamma$ is a function from
$G$ to $\Cal P(G)$ such that for each $x\in G$, $\Gamma(x)$ and
$\Gamma^{-1}\{x\}$ are finite), $K$ is a finite set of integers, and
$T$ is a function mapping subsets of $K$ into subsets of $K$.
If for each finite subgraph $(A,\Gamma_A)$ there is a function $\psi$
such that for each $x\in A$, $\psi(x)\in T(\psi[\Gamma_A(x)])$, then
there is a function $\phi$ such that for all $x\in G$, $\phi(x)\in
T(\phi[\Gamma(x)])$. \ac{Foster} \cite{1964}.
\bye