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

<b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="topgrp_1.html#M3">Homeomorphism</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="sprect_1.html#D3">SPRECT_1:def 3</a><br/>

<b>consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a> ;<br/>
<a NAME="E3:23"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E5:23"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>given </b><font color="Maroon">A1</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>, <font color="Maroon">A2</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><a NAME="E6:23"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> )
 <b>and </b><br/><a NAME="E7:23"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">A1</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">A2</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  )
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="sprect_1.html#D3">SPRECT_1:def 3</a>
<div class="add"><b>assume </b><a NAME="E1:23_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:23_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</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/>;<br/><b>then consider </b><font color="Maroon">s'</font> being   <a href="struct_0.html#NM1">Element</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="E4:23_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">s'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/><b>thus </b><a NAME="E5:23_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T77">FUNCT_2:77</a></i>;<br/></div>
<b>end;</b></div>

<b>take </b><font color="Maroon">B1</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A1</font>;<br/>

<b>take </b><font color="Maroon">B2</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A2</font>;<br/>

<a NAME="E9:23"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T153">RELAT_1:153</a></i>;<br/>
<b>hence </b><a NAME="E10:23"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="ref" href="jordan1k.html#T5">JORDAN1K:5</a></i>;<br/>

<b>thus </b><a NAME="E11:23"/>
<font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T125">FUNCT_1:125</a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B1</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B1</font>
<b>by </b><i><a class="ref" href="tops_2.html#T74">TOPS_2:74</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T123">FUNCT_1:123</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B2</font>
<b>by </b><i><a class="ref" href="funct_1.html#T123">FUNCT_1:123</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B2</font>
<b>by </b><i><a class="ref" href="tops_2.html#T74">TOPS_2:74</a></i>
;<br/>

<a NAME="E13:23"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="jordan1k.html#T5">JORDAN1K:5</a></i>;<br/>
<b>hence </b><a NAME="E14:23"/>
( <font color="Maroon">B1</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">B2</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i>, </i>;<br/>


</div>
