<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></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">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><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">E31</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><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">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><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">E33</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><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">E34</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">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:26_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><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#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E5:26"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E1:26_1"><i><font color="Green">E35</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">E35</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">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:26_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E29</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#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E5:26"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E1:26_2"><i><font color="Green">E36</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">c<sub>7</sub></font> =  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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#E1:26"><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">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E5:26"><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">E37</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></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><a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E7:26_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="jordan16.html#T9">JORDAN16:9</a></i>;<br/><a NAME="E8:26_3"/><i><font color="Green">E39</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> 
 <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/><a NAME="E9:26_3"/><i><font color="Green">E40</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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 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> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/><a NAME="E10:26_3"/><i><font color="Green">E41</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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E11:26_3"/><i><font color="Green">E42</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><a NAME="E12:26_3"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan20.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E13:26_3"/><i><font color="Green">E44</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E14:26_3"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan20.html#T23" target="_self">Th23</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan20.html#T25" target="_self">Th25</a></i>
;<br/><a NAME="E16:26_3"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan20.html#E16:26_3"><i><font color="Green">E45</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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>10</sub></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">E46</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E5:26_3_2_1_1"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26"><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">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>11</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</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">c<sub>1</sub></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#E9:26_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3_2_1_1"><i><font color="Green">E47</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">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></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">E49</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">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:26_3_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_1"><i><font color="Green">E46</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#E6:26"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E1:26_3_2_1_1_1"><i><font color="Green">E50</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">E50</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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">E51</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:26_3_2_1_1"><i><font color="Green">E49</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">E52</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26"><i><font color="Green">E34</font></i></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">E53</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26"><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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><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#E4:26"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_1"><i><font color="Green">E53</font></i></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">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="jordan20.html#T20" target="_self">Th20</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></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">E55</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></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">E56</font></i>: 
<font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green">E55</font></i></a>, <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">E57</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></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#E8:26_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E58</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green">E55</font></i></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></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">b<sub>1</sub></font> where B 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">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></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">c<sub>18</sub></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">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></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">E59</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<a NAME="E4:26_3_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_1_2_1_1_2"><i><font color="Green">E59</font></i></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">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_1_2_1_1_2"><i><font color="Green">E59</font></i></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">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></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">E59</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="struct_0.html#NM7">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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E10:26_3_2_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan20.html#E6:26_3_2_1_1_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3_2_1_1_2_1_1"><i><font color="Green">E58</font></i></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">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></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">c<sub>19</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_2_1_1_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>19</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></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">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>20</sub></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">E61</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></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">c<sub>12</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_1_2_1_1_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_1_2_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26_3_2_1_1_2_1_1"><i><font color="Green">E54</font></i></a>, <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></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#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E9:26_3_2_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E62</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></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">E63</font></i>: 
<font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E64</font></i>: <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></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">E65</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>18</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green">E55</font></i></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1_3"><i><font color="Green">E61</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">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>12</sub></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">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>18</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></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">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E9:26_3_2_1_1_2_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1_2_1_1_3"><i><font color="Green">E65</font></i></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">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26_3_2_1_1_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_1_2_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_1_2_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_1_2_1_1_3"><i><font color="Green">E64</font></i></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">E60</font></i>: 
<font color="Maroon">c<sub>18</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green">E55</font></i></a>, <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">c<sub>18</sub></font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1_2_1_1"><i><font color="Green">E60</font></i></a>, <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">b<sub>1</sub></font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1_2_1_1"><i><font color="Green">E60</font></i></a></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">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_1_2_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>13</sub></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">c<sub>17</sub></font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_1_2_1_2"><i><font color="Green">E55</font></i></a>, <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">b<sub>1</sub></font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_1_2_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_1_2_1_2"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font><b> such that </b><br/><a NAME="E18:26_3_2_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/><a NAME="E19:26_3_2_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan20.html#T22" target="_self">Th22</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E21:26_3_2_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">E57</font></i>: 
<font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green">E56</font></i></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">E58</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></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#E8:26_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E59</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>18</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green">E56</font></i></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></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">E60</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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">b<sub>1</sub></font> where B 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">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></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">c<sub>19</sub></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>20</sub></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">E61</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 ;<br/>
<a NAME="E4:26_3_2_1_1_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_1_4"><i><font color="Green">E61</font></i></a>, <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">E63</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:26_3_2_1_1_4_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</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#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_1_4"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan20.html#E1:26_3_2_1_1_4_1"><i><font color="Green">E63</font></i></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_1_4"><i><font color="Green">E62</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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></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">E61</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM7">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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E23:26_3_2_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan20.html#E24:26_3_2_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan20.html#E25:26_3_2_1_1"><i><font color="Green">E60</font></i></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">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></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">c<sub>20</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_5"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>20</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></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">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></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">E63</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></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">c<sub>12</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_1_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_1_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E19:26_3_2_1_1"><i><font color="Green">E55</font></i></a>, <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></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#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E9:26_3_2_1_1_5"/><b>then </b><i><font color="Green">E64</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></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">E65</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_5"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E66</font></i>: <font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></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">E67</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green">E56</font></i></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_5"><i><font color="Green">E63</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">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>12</sub></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">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></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">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E27:26_3_2_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1_5"><i><font color="Green">E67</font></i></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_5"/>
<font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E22:26_3_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_1_5"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_1_5"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_1_5"><i><font color="Green">E66</font></i></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">E62</font></i>: 
<font color="Maroon">c<sub>19</sub></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">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan20.html#E30:26_3_2_1_1"><i><font color="Green">E62</font></i></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">E63</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E18:26_3_2_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan20.html#E30:26_3_2_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan20.html#E34"><i><font color="Green">Lemma26</font></i></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">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_1_6_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><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#T22" target="_self">Th22</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></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">E65</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">E66</font></i>: 
<font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green">E65</font></i></a>, <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">E67</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></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#E8:26_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E68</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green">E65</font></i></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></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">E69</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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">b<sub>1</sub></font> where B 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">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></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">c<sub>22</sub></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">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>23</sub></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">E70</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<a NAME="E4:26_3_2_1_1_6_1_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E5:26"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_1_6_1_1_2"><i><font color="Green">E70</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">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_1_6_1_1_2"><i><font color="Green">E70</font></i></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_6_1_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></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">E70</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM7">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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_1_6_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3_2_1_1_6_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan20.html#E8:26_3_2_1_1_6_1_1"><i><font color="Green">E69</font></i></a>, <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">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></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">c<sub>23</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_6_1_1_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>23</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></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">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>24</sub></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">E72</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></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">c<sub>12</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_1_6_1_1_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_1_6_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26_3_2_1_1_6_1_1"><i><font color="Green">E64</font></i></a>, <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></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#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E9:26_3_2_1_1_6_1_1_3"/><b>then </b><i><font color="Green">E73</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></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">E74</font></i>: 
<font color="Maroon">c<sub>25</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E75</font></i>: <font color="Maroon">c<sub>21</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>25</sub></font> = 
<font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></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">E76</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>22</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green">E65</font></i></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>23</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></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">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1_3"><i><font color="Green">E72</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">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>12</sub></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">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></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">c<sub>22</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></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">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E10:26_3_2_1_1_6_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1_6_1_1_3"><i><font color="Green">E76</font></i></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_6_1_1_3"/>
<font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>23</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26_3_2_1_1_6_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_1_6_1_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_1_6_1_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_1_6_1_1_3"><i><font color="Green">E75</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">E71</font></i>: 
<font color="Maroon">c<sub>22</sub></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">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green">E65</font></i></a>, <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">c<sub>22</sub></font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_1_6_1_1"><i><font color="Green">E71</font></i></a>, <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">b<sub>1</sub></font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_1_6_1_1"><i><font color="Green">E71</font></i></a></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">E64</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_1_6_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></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">c<sub>15</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>21</sub></font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_1_6_1_2"><i><font color="Green">E65</font></i></a>, <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">b<sub>1</sub></font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_1_6_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_1_6_1_2"><i><font color="Green">E65</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E36:26_3_2_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 ;<br/><a NAME="E37:26_3_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>21</sub></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">c<sub>17</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E33:26_3_2_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E36:26_3_2_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E34"><i><font color="Green">Lemma26</font></i></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#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E7:26"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="jordan20.html#T18" target="_self">Th18</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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>10</sub></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">E46</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E5:26_3_2_1_2"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26"><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">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>11</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3_2_1_2"><i><font color="Green">E47</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">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></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">E49</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">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:26_3_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_2"><i><font color="Green">E46</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#E6:26"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E1:26_3_2_1_2_1"><i><font color="Green">E50</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">E50</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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">E51</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:26_3_2_1_2"><i><font color="Green">E49</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">E52</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:26"><i><font color="Green">E35</font></i></a>, <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">E53</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26"><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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3"><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">c<sub>13</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E4:26"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_2"><i><font color="Green">E53</font></i></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">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_2_2_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="jordan20.html#T22" target="_self">Th22</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></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">E55</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">E56</font></i>: 
<font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green">E55</font></i></a>, <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">E57</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></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#E8:26_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E58</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>17</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green">E55</font></i></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></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">b<sub>1</sub></font> where B 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">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></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">c<sub>18</sub></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">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></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">E59</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<a NAME="E4:26_3_2_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_2_2_1_1_2"><i><font color="Green">E59</font></i></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">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_2_2_1_1_2"><i><font color="Green">E59</font></i></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">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></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">E59</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="struct_0.html#NM7">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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E10:26_3_2_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan20.html#E6:26_3_2_1_2_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3_2_1_2_2_1_1"><i><font color="Green">E58</font></i></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">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></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">c<sub>19</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_2_1_1_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>19</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></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">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>20</sub></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">E61</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></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">c<sub>12</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_2_2_1_1_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_2_2_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26_3_2_1_2_2_1_1"><i><font color="Green">E54</font></i></a>, <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></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#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E9:26_3_2_1_2_2_1_1_3"/><b>then </b><i><font color="Green">E62</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></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">E63</font></i>: 
<font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E64</font></i>: <font color="Maroon">c<sub>17</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></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">E65</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>18</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green">E55</font></i></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1_3"><i><font color="Green">E61</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">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>12</sub></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">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>18</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></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">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E9:26_3_2_1_2_2_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2_2_1_1_3"><i><font color="Green">E65</font></i></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">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26_3_2_1_2_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_2_2_1_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_2_2_1_1_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_2_2_1_1_3"><i><font color="Green">E64</font></i></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">E60</font></i>: 
<font color="Maroon">c<sub>18</sub></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">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green">E55</font></i></a>, <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">c<sub>18</sub></font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2_2_1_1"><i><font color="Green">E60</font></i></a>, <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">b<sub>1</sub></font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2_2_1_1"><i><font color="Green">E60</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_2_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_2_2_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_2_2_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></font>,<font color="Maroon">c<sub>16</sub></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">c<sub>17</sub></font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_2_2_1_2"><i><font color="Green">E55</font></i></a>, <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">b<sub>1</sub></font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_2_2_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_2_2_1_2"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E18:26_3_2_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 ;<br/><a NAME="E19:26_3_2_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="jordan20.html#T20" target="_self">Th20</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E21:26_3_2_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></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">E57</font></i>: 
<font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green">E56</font></i></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">E58</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></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#E8:26_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E59</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>18</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green">E56</font></i></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></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">E60</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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">b<sub>1</sub></font> where B 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">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></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">c<sub>19</sub></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>20</sub></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">E61</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 ;<br/>
<a NAME="E4:26_3_2_1_2_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_2_4"><i><font color="Green">E61</font></i></a>, <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">E63</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:26_3_2_1_2_4_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</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#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_2_4"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan20.html#E1:26_3_2_1_2_4_1"><i><font color="Green">E63</font></i></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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_2_4"><i><font color="Green">E62</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">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></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">E61</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM7">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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E23:26_3_2_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan20.html#E24:26_3_2_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan20.html#E25:26_3_2_1_2"><i><font color="Green">E60</font></i></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">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></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">c<sub>20</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_5"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>20</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></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">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></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">E63</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></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">c<sub>12</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_2_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_2_5"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E19:26_3_2_1_2"><i><font color="Green">E55</font></i></a>, <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></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#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E9:26_3_2_1_2_5"/><b>then </b><i><font color="Green">E64</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></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">E65</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_5"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E66</font></i>: <font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></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">E67</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green">E56</font></i></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_5"><i><font color="Green">E63</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">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>12</sub></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">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></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">c<sub>19</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></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">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E27:26_3_2_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2_5"><i><font color="Green">E67</font></i></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_5"/>
<font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E22:26_3_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_2_5"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_2_5"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_2_5"><i><font color="Green">E66</font></i></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">E62</font></i>: 
<font color="Maroon">c<sub>19</sub></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">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan20.html#E30:26_3_2_1_2"><i><font color="Green">E62</font></i></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">E63</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E18:26_3_2_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan20.html#E30:26_3_2_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan20.html#E34"><i><font color="Green">Lemma26</font></i></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">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_2_6_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E2:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan20.html#T20" target="_self">Th20</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></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">E65</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></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">E66</font></i>: 
<font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green">E65</font></i></a>, <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">E67</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></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#E8:26_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><i><font color="Green">E68</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green">E65</font></i></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></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">E69</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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">b<sub>1</sub></font> where B 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">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></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">c<sub>22</sub></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">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>23</sub></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">E70</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<a NAME="E4:26_3_2_1_2_6_1_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E5:26"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_2_6_1_1_2"><i><font color="Green">E70</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">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_2_6_1_1_2"><i><font color="Green">E70</font></i></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_6_1_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></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">E70</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM7">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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_2_6_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3_2_1_2_6_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan20.html#E8:26_3_2_1_2_6_1_1"><i><font color="Green">E69</font></i></a>, <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">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></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">c<sub>23</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_6_1_1_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>23</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></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">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>24</sub></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">E72</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></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">c<sub>12</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_2_6_1_1_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_2_6_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26_3_2_1_2_6_1_1"><i><font color="Green">E64</font></i></a>, <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></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#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E9:26_3_2_1_2_6_1_1_3"/><b>then </b><i><font color="Green">E73</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></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">E74</font></i>: 
<font color="Maroon">c<sub>25</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<i><font color="Green">E75</font></i>: <font color="Maroon">c<sub>21</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>25</sub></font> = 
<font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></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">E76</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>22</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green">E65</font></i></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>23</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></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">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1_3"><i><font color="Green">E72</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">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>12</sub></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">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></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">c<sub>22</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></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">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E10:26_3_2_1_2_6_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2_6_1_1_3"><i><font color="Green">E76</font></i></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_6_1_1_3"/>
<font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>23</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26_3_2_1_2_6_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_2_6_1_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_2_6_1_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_2_6_1_1_3"><i><font color="Green">E75</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">E71</font></i>: 
<font color="Maroon">c<sub>22</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="borsuk_2.html#R2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green">E65</font></i></a>, <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">c<sub>22</sub></font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_2_6_1_1"><i><font color="Green">E71</font></i></a>, <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">b<sub>1</sub></font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_2_6_1_1"><i><font color="Green">E71</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_6_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_2_6_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM7">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">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_2_6_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></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">c<sub>15</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>21</sub></font> is    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_2_6_1_2"><i><font color="Green">E65</font></i></a>, <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">b<sub>1</sub></font> being   <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_2_6_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_2_6_1_2"><i><font color="Green">E65</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font><b> such that </b><br/><a NAME="E36:26_3_2_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/><a NAME="E37:26_3_2_1_2"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E33:26_3_2_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E36:26_3_2_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E34"><i><font color="Green">Lemma26</font></i></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#E1:26"><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#E8:26"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="jordan20.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E18:26_3"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:26"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E6:26"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan20.html#T21" target="_self">Th21</a></i>;<br/>


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