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<b>let </b><font color="Maroon">p</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">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">q</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="E1:37"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> )
 <b>and </b><br/><a NAME="E4:37"/><i><font color="Green">E54</font></i>: 
( ( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  ) )
 ;<br/>

<a NAME="E5:37"/><i><font color="Green">E55</font></i>: 
not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">p2</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:37"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">p1</font>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p2</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p1</font> )
 </span><b>by </b><i>, , , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>suppose </b></a><a NAME="E1:37_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T11">TOPREAL1:11</a></i>;<br/><a NAME="E3:37_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:37_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:37"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal1.html#T11">TOPREAL1:11</a></i>;<br/><b>hence </b><a NAME="E5:37_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p</font> )
 <b>by </b><i><a class="txt" href="jordan12.html#E1"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="jordan12.html#E1:37"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2"><b>suppose </b></a><a NAME="E1:37_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p1</font>
 ;<br/><div class="add"><a NAME="E2:37_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p1</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T11">TOPREAL1:11</a></i>;<br/><a NAME="E3:37_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:37_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p1</font>
 <b>by </b><i><a class="txt" href="jordan12.html#E2:37"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal1.html#T11">TOPREAL1:11</a></i>;<br/><b>hence </b><a NAME="E5:37_1_2"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p</font> or <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, <a class="txt" href="jordan12.html#E1:37"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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