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<div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</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> 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="E1:36"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E3:36"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:36"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E5:36"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font>
 <b>and </b><br/><a NAME="E6:36"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E7:36"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font>
 <b>and </b><br/><a NAME="E8:36"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 ;<br/>

<b>assume </b><a NAME="E9:36"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">A</font> being  <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>  st <font color="Olive">A</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="tarski.html#K2">}</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan18.html#D3" target="_self">JORDAN18:def 3</a><br/>

<b>consider </b><font color="Maroon">P</font> being  non <a href="xboole_0.html#V1">empty</a> <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>, <font color="Maroon">P1</font> being  non <a href="xboole_0.html#V1">empty</a> <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><b> such that </b><br/><a NAME="E11:36"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E12:36"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">P1</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E13:36"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P1</font>
 <b>and </b><br/><a NAME="E14:36"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, , , <a class="ref" href="topreal2.html#T5">TOPREAL2:5</a></i>;<br/>
<a NAME="E15:36"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E16:36"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E17:36"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1"/>
( ( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) )
 </span><b>by </b><i>, <a class="ref" href="zfmisc_1.html#T57">ZFMISC_1:57</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:36_1_1"/><i><font color="Green">E81</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="E2:36_1_1"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">A</font> =  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">q1</font>;<br/><a NAME="E1:36_1_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i>, , <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E2:36_1_1_1"/>
<font color="Maroon">A</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font>
 <b>by </b><i>, , <a class="ref" href="jordan16.html#T36">JORDAN16:36</a></i>;<br/><a NAME="E3:36_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5">JORDAN16:5</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:36_1_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Olive">q</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">q</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><a NAME="E3:36_1_1_1_1"/><b>then </b>
 ex <font color="Olive">q</font> being  <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> st <br/>( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Olive">q</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">q</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i><a class="txt" href="jordan18.html#E1"><i><font color="Green">Lemma46</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T70">JORDAN6:70</a></i>;<br/></div><b>end;</b></div><a NAME="E6:36_1_1_1"/>
not <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:36_1_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">q2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T57">ZFMISC_1:57</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:36_1_1"/>
<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="jordan18.html#NR2" target="_self">are_neighbours_wrt</a> <font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">C</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:36_1_2"/><i><font color="Green">E86</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="E2:36_1_2"/><i><font color="Green">E87</font></i>: 
not <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">A</font> =  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p1</font>,<font color="Maroon">q1</font>;<br/><a NAME="E1:36_1_2_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E2:36_1_2_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font>,<font color="Maroon">P1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:36_1_2_1"/>
<font color="Maroon">A</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font>
 <b>by </b><i>, , <a class="ref" href="jordan16.html#T36">JORDAN16:36</a></i>;<br/><a NAME="E4:36_1_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5">JORDAN16:5</a></i>;<br/><b>hence </b><a NAME="E5:36_1_2_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_2_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:36_1_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Olive">q</font>,<font color="Maroon">P1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">q</font>,<font color="Maroon">q1</font>,<font color="Maroon">P1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><a NAME="E3:36_1_2_1_1"/><b>then </b>
 ex <font color="Olive">q</font> being  <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> st <br/>( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Olive">q</font>,<font color="Maroon">P1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">q</font>,<font color="Maroon">q1</font>,<font color="Maroon">P1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i><a class="txt" href="jordan18.html#E1"><i><font color="Green">Lemma46</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_1_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T70">JORDAN6:70</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_2_1_2"/>
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:36_1_2_1_2"/><b>then </b>
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:36_1_2_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:36_1_2_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">q2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T57">ZFMISC_1:57</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:36_1_2"/>
<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="jordan18.html#NR2" target="_self">are_neighbours_wrt</a> <font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">C</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:36_1_3"/><i><font color="Green">E91</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="E2:36_1_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E3:36_1_3"/>
<font color="Maroon">P1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:36_1_3"/><b>then </b>
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> or <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T57">ZFMISC_1:57</a></i>;<br/><a NAME="E5:36_1_3"/><b>then </b>
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E6:36_1_3"/>
<font color="Maroon">p1</font>,<font color="Maroon">q1</font> <a href="jordan18.html#NR2" target="_self">are_neighbours_wrt</a> <font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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