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<div class="add">

<b>let </b><font color="Maroon">P</font> be  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">Q1</font> be  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>;<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>, <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>, <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>, <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><a NAME="E1:26"/><i><font color="Green">E27</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> &amp; <font color="Maroon">Q1</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">Q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E3:26"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan6.html#T49">JORDAN6:49</a></i>;<br/>
<a NAME="E6:26"/><b>then </b><i><font color="Green">E37</font></i>: 
 <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">q1</font>,<font color="Maroon">q2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan16.html#T36">JORDAN16:36</a></i>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 ;<br/><a NAME="E2:26_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:26_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:26"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 ;<br/><a NAME="E2:26_2"/>
 <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="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:26_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:26"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">Q0</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">q1</font>,<font color="Maroon">q2</font> as  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>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan16.html#T22">JORDAN16:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3"/>
not <font color="Maroon">Q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q0</font>
 ;<br/><b>then consider </b><font color="Maroon">x8</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> &amp; not <font color="Maroon">x8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q0</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:26_3"/>
<font color="Maroon">Q0</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:26"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan16.html#T36">JORDAN16:36</a></i>;<br/><a NAME="E5:26_3"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q0</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q0</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x8</font> as   <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> <b>by </b><i/>;<br/><a NAME="E7:26_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="jordan16.html#T9">JORDAN16:9</a></i>;<br/><a NAME="E8:26_3"/><i><font color="Green">E71</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 ;<br/><a NAME="E9:26_3"/><i><font color="Green">E73</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E10:26_3"/><i><font color="Green">E76</font></i>: 
 <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="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><a NAME="E11:26_3"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><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">q1</font>,<font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a>  <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">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, </i>;<br/><a NAME="E12:26_3"/><i><font color="Green">E79</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan20.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E13:26_3"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><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">q2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span> = 
 <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">p2</font>
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">P</font>
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, </i>
;<br/><a NAME="E15:26_3"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <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">q2</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <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">q2</font>,<font color="Maroon">p2</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_3_2_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <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> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <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">q2</font>,<font color="Maroon">p2</font> )
 </span><b>by </b><i>, , <a class="txt" href="jordan20.html#E8:26"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1"><b>case </b></a><a NAME="E1:26_3_2_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <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/><div class="add"><a NAME="E2:26_3_2_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p3</font> where <font color="Olive">p3</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">p3</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">p3</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/><b>then consider </b><font color="Maroon">p3</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><b> such that </b><br/><a NAME="E4:26_3_2_1_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</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="Maroon">p3</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/><a NAME="E5:26_3_2_1_1"/><i><font color="Green">E83</font></i>: 
not <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:26_3_2_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon">Qa</font> = <font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E3:26"><i><font color="Green">E30</font></i></a>, , <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon">Qa'</font> = <font color="Maroon">Qa</font> as   <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/><div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_2_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 ;<br/><a NAME="E2:26_3_2_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/><b>hence </b><a NAME="E3:26_3_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26"><i><font color="Green">E33</font></i></a>, , , <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/></div><b>end;</b></div><a NAME="E10:26_3_2_1_1"/><i><font color="Green">E88</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E11:26_3_2_1_1"/><i><font color="Green">E89</font></i>: 
not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E12:26_3_2_1_1"/><i><font color="Green">E91</font></i>: 
not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:26_3_2_1_1"/><i><font color="Green">E93</font></i>: 
not <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E14:26_3_2_1_1"/>
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">p1'</font> = <font color="Maroon">p1</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">q2'</font> = <font color="Maroon">q2</font>, <font color="Maroon">p2'</font> = <font color="Maroon">p2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26"><i><font color="Green">E30</font></i></a>, , , , <a class="ref" href="jordan20.html#T7" target="_self">Th7</a>, , <a class="ref" href="jordan20.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_3_2_1_1_2_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font> or <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_2_1_1"><b>case </b></a><a NAME="E1:26_3_2_1_1_2_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
 <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">q</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , </i>;<br/><b>then consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:26_3_2_1_1_2_1_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">f1</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:26_3_2_1_1_2_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:26_3_2_1_1_2_1_1"/><i><font color="Green">E97</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E99</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <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">q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E8:26_3_2_1_1_2_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">p</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">p</font>,<font color="Maroon">q</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">p</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">p</font>,<font color="Maroon">q</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</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><b> such that </b><br/><a NAME="E3:26_3_2_1_1_2_1_1_2"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p</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="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, , <a class="ref" href="jordan20.html#D5" target="_self">Def5</a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/>
<a NAME="E5:26_3_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="jordan20.html#D5" target="_self">Def5</a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:26_3_2_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:26_3_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
 <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">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">f1</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <b>by </b><i>, <a class="ref" href="jordan20.html#D3" target="_self">Def3</a>, <a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E11:26_3_2_1_1_2_1_1"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_2_1_1_3"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_1_2_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">G4</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><b> such that </b><br/><a NAME="E4:26_3_2_1_1_2_1_1_3"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">G4</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G5</font> = <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><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">q</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_1_2_1_1_3"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <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">q</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_1_2_1_1_3"/><b>then </b>
not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_1_2_1_1_3"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E109</font></i>: 
not <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><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">q</font></span>)</span> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_1_2_1_1_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">G5</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E113</font></i>: <font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G5</font> = 
<font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><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">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><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">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green">E114</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <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">q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><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">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan20.html#D5" target="_self">Def5</a>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_1_2_1_1_3"/>
<font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, , <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:26_3_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><a NAME="E13:26_3_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">p1'</font>,<font color="Maroon">q'</font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i>, <a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a></i>;<br/><a NAME="E14:26_3_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon">g1</font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">p1'</font>,<font color="Maroon">q'</font>
 <b>by </b><i>, , <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E15:26_3_2_1_1_2_1_1"/>
 ex <font color="Olive">G1</font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">p1'</font>,<font color="Maroon">q'</font> st <br/>( <font color="Olive">G1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">G1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <font color="Olive">G1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_2_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_1_2_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">g01</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_1_2_1_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">g01</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g01</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <font color="Maroon">g01</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> )
 <b>by </b><i><a class="ref" href="borsuk_2.html#T4">BORSUK_2:4</a></i>;<br/><a NAME="E4:26_3_2_1_1_2_1_2"/>
<font color="Maroon">p1'</font>,<font color="Maroon">p1'</font> <a href="borsuk_2.html#R2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">g01</font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">p1'</font>,<font color="Maroon">p1'</font>
 <b>by </b><i>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:26_3_2_1_1_2_1_2"/>
 ex <font color="Olive">G1</font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">p1'</font>,<font color="Maroon">q'</font> st <br/>( <font color="Olive">G1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">G1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <font color="Olive">G1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">G1</font> being    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">p1'</font>,<font color="Maroon">q'</font><b> such that </b><br/><a NAME="E18:26_3_2_1_1"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">G1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> )
 ;<br/><a NAME="E19:26_3_2_1_1"/><i><font color="Green">E120</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q</font>,<font color="Maroon">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="jordan20.html#E6:26"><i><font color="Green">E37</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">f2</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E21:26_3_2_1_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">f2</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><a NAME="E22:26_3_2_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E23:26_3_2_1_1"/><i><font color="Green">E123</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26"><i><font color="Green">E39</font></i></a>, <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E124</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E25:26_3_2_1_1"/><i><font color="Green">E125</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E26:26_3_2_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">q</font>,<font color="Olive">p</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">q2</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">q</font>,<font color="Olive">p</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">q2</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</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><b> such that </b><br/><a NAME="E3:26_3_2_1_1_4"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q2</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_1_4"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 
<b>by </b><i>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_2_1_1_4_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 ;<br/><a NAME="E2:26_3_2_1_1_4_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:26_3_2_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, , <a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:26_3_2_1_1_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:26_3_2_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:26_3_2_1_1"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">g2</font> = <font color="Maroon">f2</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <b>by </b><i><a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, <a class="ref" href="jordan20.html#D5" target="_self">Def5</a>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E29:26_3_2_1_1"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_5"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_1_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">G4</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><b> such that </b><br/><a NAME="E4:26_3_2_1_1_5"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">G4</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G5</font> = <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_1_5"/>
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_1_5"/><b>then </b>
not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_1_5"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_1_5"/><b>then </b><i><font color="Green">E133</font></i>: 
not <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font></span>)</span> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_1_5"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">G5</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E135</font></i>: <font color="Maroon">f2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G5</font> = 
<font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green">E136</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q</font>,<font color="Maroon">q2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_1_5"/>
<font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan20.html#D3" target="_self">Def3</a>, , , , <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:26_3_2_1_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">g2</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><a NAME="E31:26_3_2_1_1"/><b>then </b>
<font color="Maroon">q'</font>,<font color="Maroon">q2'</font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">G2</font> = <font color="Maroon">g2</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q'</font>,<font color="Maroon">q2'</font> <b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/><a NAME="E33:26_3_2_1_1"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">G1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G2</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q2'</font> )
 <b>by </b><i>, <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_3_2_1_1_6_1"/>
( <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> or <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_6_1_1"><b>case </b></a><a NAME="E1:26_3_2_1_1_6_1_1"/>
<font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_1_6_1_1"/><b>then </b><i><font color="Green">E139</font></i>: 
 <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">q2</font>,<font color="Maroon">p2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q2</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26"><i><font color="Green">E30</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">f3</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:26_3_2_1_1_6_1_1"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">f3</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:26_3_2_1_1_6_1_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">f3</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:26_3_2_1_1_6_1_1"/><i><font color="Green">E143</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E144</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <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">q2</font>,<font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E8:26_3_2_1_1_6_1_1"/><i><font color="Green">E145</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E9:26_3_2_1_1_6_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">q2</font>,<font color="Olive">p</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">p</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_6_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_6_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">q2</font>,<font color="Olive">p</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">p</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</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><b> such that </b><br/><a NAME="E3:26_3_2_1_1_6_1_1_2"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">p</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="Maroon">p</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_1_6_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:26"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/>
<a NAME="E5:26_3_2_1_1_6_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:26_3_2_1_1_6_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:26_3_2_1_1_6_1_1"/><b>then </b><i><font color="Green">E147</font></i>: 
 <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">q2</font>,<font color="Maroon">p2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">g3</font> = <font color="Maroon">f3</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <b>by </b><i>, , , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E12:26_3_2_1_1_6_1_1"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_6_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_6_1_1_3"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_1_6_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">G4</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><b> such that </b><br/><a NAME="E4:26_3_2_1_1_6_1_1_3"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">G4</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G5</font> = <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_1_6_1_1_3"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <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">q2</font>,<font color="Maroon">p2</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_1_6_1_1_3"/><b>then </b>
not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_1_6_1_1_3"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_1_6_1_1_3"/><b>then </b><i><font color="Green">E150</font></i>: 
not <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_1_6_1_1_3"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">G5</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E8:26_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E152</font></i>: <font color="Maroon">f3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G5</font> = 
<font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green">E154</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g3</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <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">q2</font>,<font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E8:26_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g3</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><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">q2</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_1_6_1_1_3"/>
<font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="txt" href="jordan20.html#E9:26_3"><i><font color="Green">E73</font></i></a>, , <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:26_3_2_1_1_6_1_1"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">g3</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><a NAME="E14:26_3_2_1_1_6_1_1"/><b>then </b>
<font color="Maroon">q2'</font>,<font color="Maroon">p2'</font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i>, <a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a></i>;<br/><a NAME="E15:26_3_2_1_1_6_1_1"/><b>then </b>
<font color="Maroon">g3</font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q2'</font>,<font color="Maroon">p2'</font>
 <b>by </b><i>, , <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E16:26_3_2_1_1_6_1_1"/>
 ex <font color="Olive">G3</font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q2'</font>,<font color="Maroon">p2'</font> st <br/>( <font color="Olive">G3</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">G3</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q2'</font> &amp; <font color="Olive">G3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_6_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_1_6_1_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">g01</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_1_6_1_2"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">g01</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g01</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q2'</font> &amp; <font color="Maroon">g01</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q2'</font> )
 <b>by </b><i><a class="ref" href="borsuk_2.html#T4">BORSUK_2:4</a></i>;<br/><a NAME="E4:26_3_2_1_1_6_1_2"/>
<font color="Maroon">q2'</font>,<font color="Maroon">q2'</font> <a href="borsuk_2.html#R2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_1_6_1_2"/><b>then </b>
<font color="Maroon">g01</font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q2'</font>,<font color="Maroon">q2'</font>
 <b>by </b><i>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:26_3_2_1_1_6_1_2"/>
 ex <font color="Olive">G3</font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q2'</font>,<font color="Maroon">p2'</font> st <br/>( <font color="Olive">G3</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">G3</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q2'</font> &amp; <font color="Olive">G3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">G3</font> being    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q2'</font>,<font color="Maroon">p2'</font><b> such that </b><br/><a NAME="E36:26_3_2_1_1"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">G3</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">G3</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q2'</font> &amp; <font color="Maroon">G3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> )
 ;<br/><a NAME="E37:26_3_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G2</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G3</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G2</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G3</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G2</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G3</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> )
 <b>by </b><i><a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, , </i>;<br/><b>hence </b><a NAME="E38:26_3_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green">E48</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <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">q2</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p3</font> where <font color="Olive">p3</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">q2</font>,<font color="Olive">p3</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">p3</font>,<font color="Maroon">p2</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/><b>then consider </b><font color="Maroon">p3</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><b> such that </b><br/><a NAME="E4:26_3_2_1_2"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">p3</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="Maroon">p3</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/><a NAME="E5:26_3_2_1_2"/><i><font color="Green">E161</font></i>: 
not <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:26_3_2_1_2"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q2</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon">Qa</font> = <font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q2</font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <b>by </b><i>, , <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon">Qa'</font> = <font color="Maroon">Qa</font> as   <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/><div><i><font color="Green">E163</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_2_1_2_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 ;<br/><a NAME="E2:26_3_2_1_2_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/><b>hence </b><a NAME="E3:26_3_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26"><i><font color="Green">E33</font></i></a>, , , <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/></div><b>end;</b></div><a NAME="E10:26_3_2_1_2"/><i><font color="Green">E165</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E11:26_3_2_1_2"/><i><font color="Green">E166</font></i>: 
not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E12:26_3_2_1_2"/><i><font color="Green">E169</font></i>: 
not <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:26_3_2_1_2"/><i><font color="Green">E170</font></i>: 
not <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E14:26_3_2_1_2"/>
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="txt" href="jordan20.html#E6:26"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">p1'</font> = <font color="Maroon">p1</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">q1'</font> = <font color="Maroon">q1</font>, <font color="Maroon">p2'</font> = <font color="Maroon">p2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , , , , <a class="ref" href="jordan20.html#T7" target="_self">Th7</a>, , <a class="ref" href="jordan20.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_3_2_1_2_2_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> or <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_2_1_1"><b>case </b></a><a NAME="E1:26_3_2_1_2_2_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_2_2_1_1"/><b>then </b><i><font color="Green">E171</font></i>: 
 <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">q</font>,<font color="Maroon">p2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , </i>;<br/><b>then consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font>,<font color="Maroon">p2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:26_3_2_1_2_2_1_1"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">f1</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:26_3_2_1_2_2_1_1"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:26_3_2_1_2_2_1_1"/><i><font color="Green">E174</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E175</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <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">q</font>,<font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E8:26_3_2_1_2_2_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">q</font>,<font color="Olive">p</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">p</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">q</font>,<font color="Olive">p</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">p</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</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><b> such that </b><br/><a NAME="E3:26_3_2_1_2_2_1_1_2"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">p</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="Maroon">p</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_2_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="jordan20.html#E6:26"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="jordan20.html#D5" target="_self">Def5</a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/>
<a NAME="E5:26_3_2_1_2_2_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q2</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="jordan20.html#D5" target="_self">Def5</a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:26_3_2_1_2_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:26_3_2_1_2_2_1_1"/><b>then </b><i><font color="Green">E177</font></i>: 
 <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">q</font>,<font color="Maroon">p2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">f1</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <b>by </b><i>, <a class="ref" href="jordan20.html#D3" target="_self">Def3</a>, <a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E11:26_3_2_1_2_2_1_1"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_2_1_1_3"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_2_2_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">G4</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><b> such that </b><br/><a NAME="E4:26_3_2_1_2_2_1_1_3"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">G4</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G5</font> = <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><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">q</font>,<font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_2_2_1_1_3"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <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">q</font>,<font color="Maroon">p2</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_2_2_1_1_3"/><b>then </b>
not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_2_2_1_1_3"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_2_2_1_1_3"/><b>then </b><i><font color="Green">E180</font></i>: 
not <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><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">q</font>,<font color="Maroon">p2</font></span>)</span> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_2_2_1_1_3"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">G5</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E182</font></i>: <font color="Maroon">f1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G5</font> = 
<font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><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">q</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><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">q</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green">E183</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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">q</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <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">q</font>,<font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><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">q</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan20.html#D5" target="_self">Def5</a>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_2_2_1_1_3"/>
<font color="Maroon">g1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, , <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:26_3_2_1_2_2_1_1"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><a NAME="E13:26_3_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon">q'</font>,<font color="Maroon">p2'</font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i>, <a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a></i>;<br/><a NAME="E14:26_3_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon">g1</font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q'</font>,<font color="Maroon">p2'</font>
 <b>by </b><i>, , <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E15:26_3_2_1_2_2_1_1"/>
 ex <font color="Olive">G1</font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q'</font>,<font color="Maroon">p2'</font> st <br/>( <font color="Olive">G1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">G1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp; <font color="Olive">G1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_2_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_2_2_1_2"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">g01</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_2_2_1_2"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">g01</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g01</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> &amp; <font color="Maroon">g01</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> )
 <b>by </b><i><a class="ref" href="borsuk_2.html#T4">BORSUK_2:4</a></i>;<br/><a NAME="E4:26_3_2_1_2_2_1_2"/>
<font color="Maroon">p2'</font>,<font color="Maroon">p2'</font> <a href="borsuk_2.html#R2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon">g01</font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">p2'</font>,<font color="Maroon">p2'</font>
 <b>by </b><i>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:26_3_2_1_2_2_1_2"/>
 ex <font color="Olive">G1</font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q'</font>,<font color="Maroon">p2'</font> st <br/>( <font color="Olive">G1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">G1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp; <font color="Olive">G1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">G1</font> being    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q'</font>,<font color="Maroon">p2'</font><b> such that </b><br/><a NAME="E18:26_3_2_1_2"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">G1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> )
 ;<br/><a NAME="E19:26_3_2_1_2"/><i><font color="Green">E188</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q1</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, </i>;<br/><b>then consider </b><font color="Maroon">f2</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E21:26_3_2_1_2"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">f2</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><a NAME="E22:26_3_2_1_2"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E23:26_3_2_1_2"/><i><font color="Green">E191</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26"><i><font color="Green">E39</font></i></a>, <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E192</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E25:26_3_2_1_2"/><i><font color="Green">E193</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E26:26_3_2_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">q1</font>,<font color="Olive">p</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">q</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">q1</font>,<font color="Olive">p</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">q</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</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><b> such that </b><br/><a NAME="E3:26_3_2_1_2_4"/><i><font color="Green">E261</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">p</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_2_4"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 
<b>by </b><i>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_2_1_2_4_1"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 ;<br/><a NAME="E2:26_3_2_1_2_4_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">q2</font>,<font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:26_3_2_1_2_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="jordan20.html#E6:26"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:26_3_2_1_2_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q2</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:26_3_2_1_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:26_3_2_1_2"/><b>then </b><i><font color="Green">E264</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">g2</font> = <font color="Maroon">f2</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <b>by </b><i><a class="ref" href="jordan20.html#D4" target="_self">Def4</a>, <a class="ref" href="jordan20.html#D5" target="_self">Def5</a>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E29:26_3_2_1_2"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_5"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">G4</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><b> such that </b><br/><a NAME="E4:26_3_2_1_2_5"/><i><font color="Green">E266</font></i>: 
( <font color="Maroon">G4</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G5</font> = <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_2_5"/>
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_2_5"/><b>then </b>
not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_2_5"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_2_5"/><b>then </b><i><font color="Green">E267</font></i>: 
not <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font></span>)</span> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_2_5"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">G5</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E269</font></i>: <font color="Maroon">f2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G5</font> = 
<font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green">E270</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">Q1</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_2_5"/>
<font color="Maroon">g2</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan20.html#D3" target="_self">Def3</a>, , , , <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:26_3_2_1_2"/><b>then </b><i><font color="Green">E271</font></i>: 
<font color="Maroon">g2</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><a NAME="E31:26_3_2_1_2"/><b>then </b>
<font color="Maroon">q1'</font>,<font color="Maroon">q'</font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, <a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">G2</font> = <font color="Maroon">g2</font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q1'</font>,<font color="Maroon">q'</font> <b>by </b><i><a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/><a NAME="E33:26_3_2_1_2"/><i><font color="Green">E272</font></i>: 
( <font color="Maroon">G2</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q1'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> )
 <b>by </b><i>, <a class="ref" href="jordan20.html#D2" target="_self">Def2</a>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_3_2_1_2_6_1"/>
( <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font> or <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_6_1_1"><b>case </b></a><a NAME="E1:26_3_2_1_2_6_1_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_2_6_1_1"/><b>then </b><i><font color="Green">E273</font></i>: 
 <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> <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="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, , </i>;<br/><b>then consider </b><font color="Maroon">f3</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:26_3_2_1_2_6_1_1"/><i><font color="Green">E274</font></i>: 
( <font color="Maroon">f3</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:26_3_2_1_2_6_1_1"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">f3</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:26_3_2_1_2_6_1_1"/><i><font color="Green">E276</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E277</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <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>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E8:26_3_2_1_2_6_1_1"/><i><font color="Green">E278</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E9:26_3_2_1_2_6_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">p</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">p</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_6_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_6_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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">p</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">p</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</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><b> such that </b><br/><a NAME="E3:26_3_2_1_2_6_1_1_2"/><i><font color="Green">E279</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p</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="Maroon">p</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_2_6_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:26"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/>
<a NAME="E5:26_3_2_1_2_6_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q2</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:26_3_2_1_2_6_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:26_3_2_1_2_6_1_1"/><b>then </b><i><font color="Green">E280</font></i>: 
 <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> <a href="tarski.html#R1">c=</a> <font color="Maroon">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">g3</font> = <font color="Maroon">f3</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span> <b>by </b><i>, , , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E12:26_3_2_1_2_6_1_1"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_6_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_6_1_1_3"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_2_6_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">G4</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><b> such that </b><br/><a NAME="E4:26_3_2_1_2_6_1_1_3"/><i><font color="Green">E282</font></i>: 
( <font color="Maroon">G4</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G5</font> = <font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><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></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_2_6_1_1_3"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_2_6_1_1_3"/><b>then </b>
not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_2_6_1_1_3"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_2_6_1_1_3"/><b>then </b><i><font color="Green">E283</font></i>: 
not <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><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></span>)</span> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_2_6_1_1_3"/><i><font color="Green">E284</font></i>: 
<font color="Maroon">G5</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E8:26_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E285</font></i>: <font color="Maroon">f3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G5</font> = 
<font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">G4</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green">E286</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g3</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <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>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E8:26_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g3</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G4</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_2_6_1_1_3"/>
<font color="Maroon">g3</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="txt" href="jordan20.html#E9:26_3"><i><font color="Green">E73</font></i></a>, , <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:26_3_2_1_2_6_1_1"/><b>then </b><i><font color="Green">E287</font></i>: 
<font color="Maroon">g3</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><a NAME="E14:26_3_2_1_2_6_1_1"/><b>then </b>
<font color="Maroon">p1'</font>,<font color="Maroon">q1'</font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i>, <a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a></i>;<br/><a NAME="E15:26_3_2_1_2_6_1_1"/><b>then </b>
<font color="Maroon">g3</font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">p1'</font>,<font color="Maroon">q1'</font>
 <b>by </b><i>, , <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E16:26_3_2_1_2_6_1_1"/>
 ex <font color="Olive">G3</font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">p1'</font>,<font color="Maroon">q1'</font> st <br/>( <font color="Olive">G3</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">G3</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <font color="Olive">G3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <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="26_3_2_1_2_6_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_2_6_1_2"/><i><font color="Green">E288</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">g01</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Qa</font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_2_6_1_2"/><i><font color="Green">E289</font></i>: 
( <font color="Maroon">g01</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g01</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q1'</font> &amp; <font color="Maroon">g01</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q1'</font> )
 <b>by </b><i><a class="ref" href="borsuk_2.html#T4">BORSUK_2:4</a></i>;<br/><a NAME="E4:26_3_2_1_2_6_1_2"/>
<font color="Maroon">q1'</font>,<font color="Maroon">q1'</font> <a href="borsuk_2.html#R2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_2_6_1_2"/><b>then </b>
<font color="Maroon">g01</font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">q1'</font>,<font color="Maroon">q1'</font>
 <b>by </b><i>, <a class="ref" href="borsuk_2.html#D2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:26_3_2_1_2_6_1_2"/>
 ex <font color="Olive">G3</font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">p1'</font>,<font color="Maroon">q1'</font> st <br/>( <font color="Olive">G3</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">G3</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <font color="Olive">G3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q1'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">G3</font> being    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">p1'</font>,<font color="Maroon">q1'</font><b> such that </b><br/><a NAME="E36:26_3_2_1_2"/><i><font color="Green">E290</font></i>: 
( <font color="Maroon">G3</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">G3</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <font color="Maroon">G3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q1'</font> )
 ;<br/><a NAME="E37:26_3_2_1_2"/>
( <font color="Maroon">G3</font> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G1</font></span>)</span> is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">G3</font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">G3</font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2'</font> )
 <b>by </b><i><a class="ref" href="jordan20.html#D6" target="_self">Def6</a>, , </i>;<br/><b>hence </b><a NAME="E38:26_3_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26"><i><font color="Green">E30</font></i></a>, , <a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green">E48</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:26_3"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:26"/>
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a>  <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">q1</font>,<font color="Maroon">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E5:26"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan20.html#T1" target="_self">Th1</a></i>;<br/>


</div>
