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

<b>let </b><font color="Maroon">P</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>, <font color="Maroon">Q</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">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>;<br/>



<b>assume </b><a NAME="E1:5"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:5"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E3:5"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:5"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:5"/>
for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span><br/> for <font color="Olive">s2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2</font> &amp; <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 holds <br/>for <font color="Olive">t</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s2</font> holds <br/> not <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<b>then </b> <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 

;<br/>
<a NAME="E3:5_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E4:5_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">g</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/>
<b>let </b><font color="Maroon">t</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E5:5_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s2</font> )
 ;<br/>

<b>hence </b><a NAME="E6:5_1"/>
not <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:5"/>
 <a href="jordan5c.html#K1" target="_self">First_Point</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>by </b><i>, , , </i>;<br/>


</div>
