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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ;<br/>





<b>assume </b><a NAME="E1:62"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">Y</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="borsuk_5.html#T62" title="BORSUK_5:th.62">BORSUK_5:62</a></i>;<br/>
<a NAME="E3:62"/><i><font color="Green" title="E46">A2</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<b>thus </b><a NAME="E4:62"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  ;<br/>
<div><i><font color="Green" title="E48">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1_1"/><i><font color="Green" title="E49">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 ;<br/><a NAME="E2:62_1_1"/><i><font color="Green" title="E50">A6</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T73" title="TOPS_1:th.73">TOPS_1:73</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font>
 ;<br/><a NAME="E2:62_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> 
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:62"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="rcomp_2.html#T9" title="RCOMP_2:th.9">RCOMP_2:9</a></i>;<br/><b>hence </b><a NAME="E3:62_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:62_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="tops_1.html#T47" title="TOPS_1:th.47">TOPS_1:47</a></i>;<br/></div><b>end;</b></div><a NAME="E4:62_1_1"/><b>then </b>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:62"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="rcomp_3.html#T46" target="_self" title="RCOMP_3:th.46">Th46</a></i>;<br/><a NAME="E5:62_1_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E6:62_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E2:62_1_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="rcomp_3.html#E1:62_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="rcomp_3.html#E1:62_1_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:62_1"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:62"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="rcomp_3.html#E1:62_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="rcomp_3.html#E3:62"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a></i>;<br/>
<a NAME="E5:62_1"/><b>then </b>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E3:62_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:62_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>


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

<a NAME="E5:62"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 
<b>by </b><i><a class="ref" href="topreala.html#T20" title="TOPREALA:th.20">TOPREALA:20</a></i>;<br/>
<a NAME="E6:62"/><b>then </b>
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:62"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="tops_1.html#T56" title="TOPS_1:th.56">TOPS_1:56</a></i>;<br/>
<b>hence </b><a NAME="E7:62"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">X</font>
 ;<br/>


</div>
