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<div class="add">

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="kurato_1.html#K6" target="_self">KurExSet</a> ;<br/>
<b>assume </b><a NAME="E1:52_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  <a href="hidden.html#R2">in</a>  <a href="kurato_1.html#K3" target="_self">Kurat14ClPart</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_5.html#D6">BORSUK_5:def 6</a><br/>

<a NAME="E2:52_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K4">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p5">(<span class="default"><a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span><a href="enumset1.html#K4">}</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_1"/>
(  <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a>  or  <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> or  <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> or  <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> </span>)</span> or  <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> or  <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p5">(<span class="default"><a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> )
 </span><b>by </b><i><a class="txt" href="kurato_1.html#E2:52_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="enumset1.html#D4">ENUMSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_1_1"/>
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="kurato_1.html#T12" target="_self">Th12</a>, <a class="ref" href="borsuk_5.html#T103">BORSUK_5:103</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_1_2"/>
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="kurato_1.html#T14" target="_self">Th14</a>, <a class="ref" href="borsuk_5.html#T73">BORSUK_5:73</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_3"><b>suppose </b></a><a NAME="E1:52_1_1_1_3"/>
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="kurato_1.html#T16" target="_self">Th16</a>, <a class="ref" href="borsuk_5.html#T72">BORSUK_5:72</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_4"><b>suppose </b></a><a NAME="E1:52_1_1_1_4"/>
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="kurato_1.html#T19" target="_self">Th19</a>, <a class="ref" href="borsuk_5.html#T104">BORSUK_5:104</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_5"><b>suppose </b></a><a NAME="E1:52_1_1_1_5"/>
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_1_5"/>
contradiction
 <b>by </b><i><a class="ref" href="kurato_1.html#T21" target="_self">Th21</a>, <a class="ref" href="borsuk_5.html#T72">BORSUK_5:72</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_6"><b>suppose </b></a><a NAME="E1:52_1_1_1_6"/>
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p5">(<span class="default"><a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_1_6"/>
contradiction
 <b>by </b><i><a class="ref" href="kurato_1.html#T23" target="_self">Th23</a>, <a class="ref" href="borsuk_5.html#T73">BORSUK_5:73</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
