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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E9">Th6</font></span>: <a NAME="T6"><span class="comment"><font color="firebrick">:: SURREAL0:6</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">O</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><br/>  for <font color="Olive" title="b2">R</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><br/>  for <font color="Olive" title="b3">L</font> being   <a href="ordinal1.html#NM4" title="ORDINAL1:NM.4">Sequence</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">O</font> &amp; (  for <font color="Olive" title="b4">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b4">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">O</font> holds <br/><font color="Olive" title="b3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b5">x</font> where <font color="Olive" title="b5">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="surreal0.html#K3" title="SURREAL0:func.3">Games</a> <font color="Olive" title="b4">A</font> : (  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Olive" title="b5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">L</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Olive" title="b4">A</font></span>)</span></span>)</span> &amp;  <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Olive" title="b5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">L</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Olive" title="b4">A</font></span>)</span></span>)</span> &amp;  <a href="surreal0.html#K1" title="SURREAL0:func.1">L_</a> <font color="Olive" title="b5">x</font> <a href="surreal0.html#R3" title="SURREAL0:pred.3">&lt;&lt;</a> <font color="Olive" title="b2">R</font>, <a href="surreal0.html#K2" title="SURREAL0:func.2">R_</a> <font color="Olive" title="b5">x</font> ) </span> } </span>  ) holds <br/> for <font color="Olive" title="b4">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b4">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">O</font> holds <br/><font color="Olive" title="b3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K5" title="SURREAL0:func.5">Day</a> (<font color="Olive" title="b2">R</font>,<font color="Olive" title="b4">A</font>)</div></div>
