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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E41">Th31</font></span>: <a NAME="T31"><span class="comment"><font color="firebrick">:: SURREAL0:31</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">X</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b3">X</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b3">X</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">B</font> holds <br/><span class="p1">(<span class="default"><a href="surreal0.html#K9" title="SURREAL0:func.9">No_Ord</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K4" title="SURREAL0:func.4">BeforeGames</a> <font color="Olive" title="b3">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="surreal0.html#K4" title="SURREAL0:func.4">BeforeGames</a> <font color="Olive" title="b3">X</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="surreal0.html#K9" title="SURREAL0:func.9">No_Ord</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="surreal0.html#K4" title="SURREAL0:func.4">BeforeGames</a> <font color="Olive" title="b3">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="surreal0.html#K4" title="SURREAL0:func.4">BeforeGames</a> <font color="Olive" title="b3">X</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></div></div>
