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<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> 2</span>)</span>;<br/><b>let </b><font color="Maroon">P1</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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</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">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q0</font> where <font color="Olive">q0</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> :  ex <font color="Olive">ss</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">ss</font> &amp; <font color="Olive">ss</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Olive">q0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">ss</font> ) </span>}</span>  )
 ;<br/>

<a NAME="E2:16"/>
<span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">s</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:16_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">s</font></span><a href="rcomp_2.html#K1">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">sx</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:16_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sx</font> &amp; <font color="Maroon">sx</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E4:16_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sx</font> &amp; <font color="Maroon">sx</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Q</font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">s</font></span><a href="rcomp_2.html#K1">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E4:16"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:16"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:16"/>
( <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"><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> )
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:16"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="tops_2.html#T64">TOPS_2:64</a></i>;<br/>
<a NAME="E8:16"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T70">TOPS_2:70</a></i>;<br/>
<a NAME="E9:16"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <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="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E10:16"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  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="E11:16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="jordan20.html#T1" target="_self">Th1</a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E12:16"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>;<br/>
<a NAME="E13:16"/>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  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>hence </b><a NAME="E14:16"/>
<font color="Maroon">P1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , , , <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


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