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

<b>let </b><font color="Maroon">f</font> be   <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/>

<b>assume </b><a NAME="E1:6"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; ( for <font color="Olive">p2</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>  ex <font color="Olive">K</font> being 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> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">K</font> &amp;  ex <font color="Olive">V2</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> st <br/>( <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">V2</font> &amp; <font color="Olive">V2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V2</font> <a href="tarski.html#R1">c=</a> <font color="Olive">K</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">V2</font> ) ) ) )
 ;<br/>

<a NAME="E2:6"/><i><font color="Green">E36</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="E3:6"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  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> <b>by </b><i>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/>
<a NAME="E5:6"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><br/> for <font color="Olive">V</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>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">W</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> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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">V</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>assume </b><a NAME="E1:6_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<b>consider </b><font color="Maroon">K</font> being  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><b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font> &amp;  ex <font color="Olive">V2</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> st <br/>( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V2</font> &amp; <font color="Olive">V2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">V2</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">V2</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="E5:6_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> &amp; <font color="Maroon">V2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:6_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> &amp; <font color="Maroon">V2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>
<i><font color="Green">E43</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
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> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
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">K</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E8:6_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font></span>)</span> <a href="tarski.html#R1">c=</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>
 
;<br/>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font> as   <a href="struct_0.html#NM6">Function</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">K</font></span>)</span>,<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#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E10:6_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/>
<a NAME="E11:6_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<i><font color="Green">E55</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">K</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<i><font color="Green">E56</font></i>: <font color="Maroon">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font>
<b>by </b><i>, <a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>
;<br/>
<b>then consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM6">Function</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">K</font></span>)</span>,<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">K</font></span>)</span><b> such that </b><br/><a NAME="E15:6_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 <b>by </b><i>, , <a class="ref" href="jgraph_1.html#T64">JGRAPH_1:64</a></i>;<br/>
<a NAME="E16:6_1"/><i><font color="Green">E59</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <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">K</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E17:6_1"/>
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">K</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R</font> = <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K</font> 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> <font color="Maroon">K</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E19:6_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E20:6_1"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font></span>)</span> <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">K</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E21:6_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">R</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<a NAME="E22:6_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E23:6_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>
<i><font color="Green">E65</font></i>: <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
;<br/>
<i><font color="Green">E66</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">K</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E26:6_1"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E27:6_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E28:6_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E29:6_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , , <a class="ref" href="jgraph_4.html#T3" target="_self">Th3</a>, , <a class="ref" href="jgraph_4.html#T4" target="_self">Th4</a>, <a class="ref" href="jgraph_4.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</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">K</font></span>)</span> <b>by </b><i>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E31:6_1"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_4.html#E2"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="jgraph_4.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E32:6_1"/><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<a NAME="E33:6_1"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E34:6_1"/><i><font color="Green">E72</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <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">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E35:6_1"/><i><font color="Green">E73</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <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">K</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM6">Function</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">K</font></span>)</span>,<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">K</font></span>)</span> <b>by </b><i>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E37:6_1"/>
<font color="Maroon">g1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f1</font> <a href="tops_2.html#NK3" target="_self">"</a> 
 
<b>by </b><i>, , , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E38:6_1"/><b>then </b>
<font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">W3</font> being   <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">K</font></span>)</span><b> such that </b><br/><a NAME="E40:6_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font> &amp; <font color="Maroon">W3</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">W3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="txt" href="jgraph_4.html#E1"><i><font color="Green">Lemma24</font></i></a>, , <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E41:6_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">W3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:6_1_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">W3</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_6"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:6_1_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_4.html#E3:6"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E5:6_1_6"/><b>then </b><i><font color="Green">E80</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="txt" href="jgraph_4.html#E3:6"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<a NAME="E6:6_1_6"/><i><font color="Green">E81</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </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">K</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E7:6_1_6"/>
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">K</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:6_1_6"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z2</font> )
 <b>by </b><i>, <a class="txt" href="jgraph_4.html#E3:6"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jgraph_4.html#E1:6_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E10:6_1_6"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_4.html#E1:6_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jgraph_4.html#E5:6_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E11:6_1_6"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E12:6_1_6"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_4.html#E1:6_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jgraph_4.html#E6:6_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E13:6_1_6"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , <a class="txt" href="jgraph_4.html#E1:6"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jgraph_4.html#E3:6"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jgraph_4.html#E3:6_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jgraph_4.html#E7:6_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<a NAME="E14:6_1_6"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">W3</font>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E3:6"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E15:6_1_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="jgraph_4.html#E2:6"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">W5</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="E43:6_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">W5</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">W3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W5</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">K</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_4.html#E2:6"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W4</font> = <font color="Maroon">W5</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</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/>
<a NAME="E45:6_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W5</font>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E2:6"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jgraph_4.html#E3:6"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E46:6_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E47:6_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">W4</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="txt" href="jgraph_4.html#E3:6"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<font color="Maroon">W4</font> = 
<font color="Maroon">W5</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">V2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W5</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">W3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V2</font>
<b>by </b><i><a class="txt" href="jgraph_4.html#E3:6"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E49:6_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W4</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W3</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T154">RELAT_1:154</a></i>;<br/>
<a NAME="E50:6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W3</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W3</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E51:6_1"/><b>then </b>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W4</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W3</font>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E5:6_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E52:6_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W4</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E53:6_1"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">V2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E54:6_1"/><b>then </b>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E55:6_1"/><b>then </b>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W4</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E6:6_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E56:6_1"/>
 ex <font color="Olive">W</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> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="txt" href="jgraph_4.html#E1:6_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jgraph_4.html#E3:6_1"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E7:6"/>
<font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> 
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:6"/>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, , , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>


</div>
