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

<b>set </b><font color="Maroon">GG</font> = <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/>
<b>set </b><font color="Maroon">SS</font> =  <a href="euclid.html#K15">TOP-REAL</a> 2;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <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>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">K</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>reconsider </b><font color="Maroon">L</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</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> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>] </span>}</span>  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> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:30"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E5:30"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, <a class="ref" href="topreal6.html#T85">TOPREAL6:85</a></i>;<br/>
<a NAME="E6:30"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_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="E1:30_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:30_1_1"/><i><font color="Green">E34</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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E5:30_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>] )
 <b>by </b><i/>;<br/>
<a NAME="E6:30_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E7:30_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:30_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">y1</font> = <font color="Maroon">y1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E11:30_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E12:30_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="euclid.html#K23">]|</a></span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <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/>
<i><font color="Green">E62</font></i>: <font color="Maroon">x'</font> <a href="euclid.html#K21">`1</a>  = 
<font color="Maroon">x'</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="euclid.html#D14">EUCLID:def 14</a></i>
<br/>.= 
<font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<i><font color="Green">E63</font></i>: <font color="Maroon">x'</font> <a href="euclid.html#K22">`2</a>  = 
<font color="Maroon">x'</font> <a href="funct_1.html#K1">.</a> 2
<b>by </b><i><a class="ref" href="euclid.html#D15">EUCLID:def 15</a></i>
<br/>.= 
<font color="Maroon">y1</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E16:30_1_1"/>
( <font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E17:30_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:30_1"/>
<font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><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="E1:30_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</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><b> such that </b><br/><a NAME="E3:30_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">z</font>] )
 ;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">y1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:30_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T55">EUCLID:55</a></i>;<br/>
<a NAME="E6:30_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:30_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E8:30_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">x1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E9:30_1_2"/><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:30_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E12:30_1_2"/>
( <font color="Maroon">Y</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">Y</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E13:30_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:30_1_2"/><i><font color="Green">E93</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>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:30_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E16:30_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E4:30"><i><font color="Green">E32</font></i></a>, <a class="txt" href="borsuk_6.html#E5:30"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:30"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span> : <font color="Olive">p</font> <a href="mcart_1.html#K2">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="mcart_1.html#K1">`1</a> </span>)</span></span>)</span> </span>}</span>  is  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="tops_2.html#T72">TOPS_2:72</a></i>;<br/>


</div>
