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<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="kurato_1.html#K6" target="_self">KurExSet</a> ,<span class="p2">(<span class="default"><a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="tarski.html#K2">}</a></span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="measure1.html#T7">MEASURE1:7</a></i>;<br/>
<a NAME="E2:53_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="borsuk_5.html#V1">with_proper_subsets</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1_1_1"/><i><font color="Green">E43</font></i>: 
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> <font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_5.html#D6">BORSUK_5:def 6</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_1_1"/>
(  <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</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="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a>  )
 </span><b>by </b><i><a class="txt" href="kurato_1.html#E1:53_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_1"/>
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="kurato_1.html#K6" target="_self">KurExSet</a> 
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_1_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a>  <a href="hidden.html#R1">=</a>  <a href="kurato_1.html#K6" target="_self">KurExSet</a> 
 <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>hence </b><a NAME="E3:53_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_2"/>
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> 
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_1_2"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a>  <a href="hidden.html#R1">=</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>hence </b><a NAME="E3:53_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:53_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="kurato_1.html#K3" target="_self">Kurat14ClPart</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> </span>)</span> is <a href="borsuk_5.html#V1">with_proper_subsets</a>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T115">BORSUK_5:115</a></i>;<br/>
<a NAME="E4:53_1_1"/>
 <a href="kurato_1.html#K2" target="_self">Kurat14Set</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="kurato_1.html#K3" target="_self">Kurat14ClPart</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="kurato_1.html#K4" target="_self">Kurat14OpPart</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="kurato_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E5:53_1_1"/>
 <a href="kurato_1.html#K2" target="_self">Kurat14Set</a> <a href="kurato_1.html#K6" target="_self">KurExSet</a>  is <a href="borsuk_5.html#V1">with_proper_subsets</a>
 <b>by </b><i><a class="txt" href="kurato_1.html#E3:53_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="borsuk_5.html#T115">BORSUK_5:115</a></i>;<br/>


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