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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="topreal1.html#R1" target="_self">is_an_arc_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:21_1_1"/><b>then </b>
 <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="topreal1.html#R1" target="_self">is_an_arc_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="topreal1.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="topreal1.html#R1" target="_self">is_an_arc_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topreal1.html#E1:21"><i><font color="Green">E14</font></i></a>, <a class="txt" href="topreal1.html#E2:21"><i><font color="Green">E15</font></i></a>, <a class="ref" href="topreal1.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:21_1_2"/><b>then </b>
(  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="topreal1.html#E1:21"><i><font color="Green">E14</font></i></a>, <a class="ref" href="topreal1.html#T4" target="_self">Th4</a>, <a class="ref" href="topreal1.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E3:21_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="topreal1.html#R1" target="_self">is_an_arc_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topreal1.html#E1:21"><i><font color="Green">E14</font></i></a>, <a class="txt" href="topreal1.html#E1:21_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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