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<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:84"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E2:84"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E3:84"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E4:84"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font>
 ;<br/>

<a NAME="E5:84"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">p2</font>,<font color="Maroon">p1</font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self">Th18</a>, <a class="ref" href="sppol_2.html#T13" target="_self">Th13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> or <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font> or <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1"><b>suppose </b></a><a NAME="E1:84_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_1_1"/>
<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self">Th18</a>, <a class="ref" href="sppol_2.html#T19" target="_self">Th19</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_2"><b>suppose </b></a><a NAME="E1:84_1_2"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_1_2"/>
<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="sppol_2.html#T20" target="_self">Th20</a>, <a class="ref" href="sppol_2.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_3"><b>suppose </b></a><a NAME="E1:84_1_3"/>
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font>
 ;<br/><div class="add"><a NAME="E2:84_1_3"/><b>then </b>
<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self">Th18</a>, , </i>;<br/><a NAME="E3:84_1_3"/><b>then </b>
<font color="Maroon">q2</font>,<font color="Maroon">q1</font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T19" target="_self">Th19</a>, , <a class="ref" href="sppol_2.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E4:84_1_3"/>
<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_4"><b>suppose </b></a><a NAME="E1:84_1_4"/>
<font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font>
 ;<br/><div class="add"><a NAME="E2:84_1_4"/><b>then </b>
<font color="Maroon">q1</font>,<font color="Maroon">p2</font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self">Th18</a>, , <a class="ref" href="sppol_2.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E3:84_1_4"/>
<font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T19" target="_self">Th19</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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