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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E27">Th21</font></span>: <a NAME="T21"><span class="comment"><font color="firebrick">:: SURREAL0:21</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><br/>  for <font color="Olive" title="b3">R</font>, <font color="Olive" title="b4">S</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>  st <font color="Olive" title="b3">R</font> is  <a href="surreal0.html#V1" title="SURREAL0:attr.1">almost-No-order</a>  &amp; <font color="Olive" title="b4">S</font> is  <a href="surreal0.html#V1" title="SURREAL0:attr.1">almost-No-order</a>  &amp; <font color="Olive" title="b3">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="surreal0.html#K7" title="SURREAL0:func.7">OpenProd</a> (<font color="Olive" title="b3">R</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="surreal0.html#K7" title="SURREAL0:func.7">OpenProd</a> (<font color="Olive" title="b4">S</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>)</span>)</span> &amp; <font color="Olive" title="b3">R</font> <a href="surreal0.html#R4" title="SURREAL0:pred.4">preserves_No_Comparison_on</a>  <a href="surreal0.html#K8" title="SURREAL0:func.8">ClosedProd</a> (<font color="Olive" title="b3">R</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) &amp; <font color="Olive" title="b4">S</font> <a href="surreal0.html#R4" title="SURREAL0:pred.4">preserves_No_Comparison_on</a>  <a href="surreal0.html#K8" title="SURREAL0:func.8">ClosedProd</a> (<font color="Olive" title="b4">S</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) holds <br/><font color="Olive" title="b3">R</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="surreal0.html#K8" title="SURREAL0:func.8">ClosedProd</a> (<font color="Olive" title="b3">R</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="surreal0.html#K8" title="SURREAL0:func.8">ClosedProd</a> (<font color="Olive" title="b4">S</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>)</span>)</span></div></div>
