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<b>let </b><font color="Maroon" title="c1">s1</font>, <font color="Maroon" title="c2">t1</font>, <font color="Maroon" title="c3">s2</font>, <font color="Maroon" title="c4">t2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">qc</font> where <font color="Olive" title="b1">qc</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qc</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or  not <font color="Olive" title="b1">qc</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> or  not <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qc</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or  not <font color="Olive" title="b1">qc</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span> 
 ;<br/>

<a NAME="E3:45"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">sa</font>,<font color="Olive" title="b2">ta</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">sa</font>, <font color="Olive" title="b2">ta</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">sa</font> &amp; <font color="Olive" title="b1">sa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">ta</font> &amp; <font color="Olive" title="b2">ta</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E1:45"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="jordan1.html#T32" target="_self" title="JORDAN1:th.32">Th32</a></i>;<br/>
<a NAME="E4:45"/>
<font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">sb</font>,<font color="Olive" title="b2">tb</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">sb</font>, <font color="Olive" title="b2">tb</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">sb</font> or  not <font color="Olive" title="b1">sb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s2</font> or  not <font color="Maroon" title="c2">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">tb</font> or  not <font color="Olive" title="b2">tb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan1.html#E2:45"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="jordan1.html#T33" target="_self" title="JORDAN1:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E5:45"/>
<font color="Maroon" title="c5">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="txt" href="jordan1.html#E3:45"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="jordan1.html#T31" target="_self" title="JORDAN1:th.31">Th31</a></i>;<br/>


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