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

<b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p3</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p4</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</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>assume </b><a NAME="E1:70"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p3</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p4</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">q1</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>, <font color="Maroon">q2</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>, <font color="Maroon">q3</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>, <font color="Maroon">q4</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="E3:70"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds  <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> ) &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font> &amp; <font color="Maroon">q4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> &amp; <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">q3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">q3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">q4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">q4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">q3</font>,<font color="Maroon">q4</font>,<font color="Maroon">P</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:70"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</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="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:70"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E6:70"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E7:70"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q3</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E8:70"/>
<font color="Maroon">q3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q4</font>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">f2</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E10:70"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">f2</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds  <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> ) &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q1</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q2</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q3</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q4</font> )
 <b>by </b><i><a class="ref" href="jgraph_5.html#T11" target="_self">Th11</a>, , , , </i>;<br/>
<b>reconsider </b><font color="Maroon">f3</font> = <font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E12:70"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f3</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E15:70"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<a NAME="E13:70"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:70"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds  <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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/>

<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E16:70"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E15:70"><i><font color="Green">E73</font></i></a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:70_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:70"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">f3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E15:70"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E16:70"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E16:70"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">f3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E15:70"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E16:70"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E17:70"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">f3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E15:70"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E16:70"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E18:70"/>
<font color="Maroon">f3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E15:70"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E16:70"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E19:70"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds  <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> ) &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> )
 <b>by </b><i>, , <a class="txt" href="jgraph_5.html#E17:70"><i><font color="Green">E77</font></i></a>, , </i>;<br/>


</div>
