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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<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> <font color="Maroon">n</font></span>)</span>, <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> <font color="Maroon">n</font></span>)</span>, <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> <font color="Maroon">n</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> or <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font> )
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p2</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> or <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>suppose </b></a><a NAME="E1:23_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1"/><b>then </b>
 <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="topreal1.html#R1" target="_self">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:23_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</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" target="_self">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">q1</font></span>)</span> <a href="topreal1.html#R1" target="_self">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><b>suppose </b></a><a NAME="E1:23_1_2"/>
<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="E2:23_1_2"/><b>then </b>
 <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">q1</font> <a href="topreal1.html#R1" target="_self">is_an_arc_of</a> <font color="Maroon">p2</font>,<font color="Maroon">q1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:23_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</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" target="_self">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">q1</font></span>)</span> <a href="topreal1.html#R1" target="_self">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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