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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</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> <font color="Maroon">n</font></span>)</span>;<br/><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"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green">E29</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E3:20"/>
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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E4:20"/><b>then </b><i><font color="Green">E30</font></i>: 
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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</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="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">g</font> as   <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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E6:20"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 ;<br/>

<i><font color="Green">E36</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> = 
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="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <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="E8:20"/><i><font color="Green">E37</font></i>: 
 <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"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E9:20"/><i><font color="Green">E38</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</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="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E10:20"/>
for <font color="Olive">P2</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>  st <font color="Olive">P2</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">P2</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P2</font> be   <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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:20_2"/>
<font color="Maroon">P2</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>then consider </b><font color="Maroon">C</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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:20_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:20_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">C</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<a NAME="E5:20_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T14">PRE_TOPC:14</a></i>;<br/>
<a NAME="E6:20_2"/>
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i>, , <a class="ref" href="relset_1.html#T38">RELSET_1:38</a></i>;<br/>
<b>then </b><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>
<b>by </b><i><a class="txt" href="jordan7.html#E1"><i><font color="Green">Lemma23</font></i></a>, , <a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">C</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E8:20_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P2</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:20"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</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="E12:20"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T3">TOPMETR:3</a></i>;<br/>
<b>hence </b><a NAME="E13:20"/>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, , , <a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="compts_1.html#T26">COMPTS_1:26</a></i>;<br/>


</div>
