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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E30">Th24</font></span>: <a NAME="T24"><span class="comment"><font color="firebrick">:: SURREAL0:24</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">Lp</font>, <font color="Olive" title="b2">Lr</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="b1">Lp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b2">Lr</font> &amp; (  for <font color="Olive" title="b3">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b3">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">Lp</font> holds <br/>(  ex <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b6">R</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Olive" title="b6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Lr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b1">Lp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreal0.html#K8" title="SURREAL0:func.8">ClosedProd</a> (<font color="Olive" title="b6">R</font>,<font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font>) ) &amp; <font color="Olive" title="b2">Lr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">A</font> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> &amp; (  for <font color="Olive" title="b4">R</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>  st <font color="Olive" title="b4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Lr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">A</font> holds <br/>( <font color="Olive" title="b4">R</font> <a href="surreal0.html#R4" title="SURREAL0:pred.4">preserves_No_Comparison_on</a> <font color="Olive" title="b1">Lp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b4">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Lp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">A</font> ) ) ) ) holds <br/>(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b2">Lr</font></span>)</span> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> &amp; (  for <font color="Olive" title="b3">R</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>  st <font color="Olive" title="b3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b2">Lr</font></span>)</span> holds <br/>( <font color="Olive" title="b3">R</font> <a href="surreal0.html#R4" title="SURREAL0:pred.4">preserves_No_Comparison_on</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">Lp</font></span>)</span> &amp; <font color="Olive" title="b3">R</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="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">Lp</font></span>)</span> &amp; (  for <font color="Olive" title="b4">A</font>, <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><br/>  for <font color="Olive" title="b7">S</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>  st <font color="Olive" title="b4">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">Lp</font> &amp; <font color="Olive" title="b7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Lr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">A</font> &amp; <font color="Olive" title="b1">Lp</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#K8" title="SURREAL0:func.8">ClosedProd</a> (<font color="Olive" title="b7">S</font>,<font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font>) holds <br/><font color="Olive" title="b3">R</font> <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="b5">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="surreal0.html#K4" title="SURREAL0:func.4">BeforeGames</a> <font color="Olive" title="b5">a</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> <font color="Olive" title="b7">S</font> <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="b5">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="surreal0.html#K4" title="SURREAL0:func.4">BeforeGames</a> <font color="Olive" title="b5">a</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) ) ) )</div></div>
