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<b>take </b>
 <a href="topmetr.html#K3">R^1</a> 
;<br/>

<b>set </b><font color="Maroon">B</font> =  <a href="borsuk_5.html#NK2" target="_self">].-infty,</a> 1;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="borsuk_5.html#K2">RAT</a> 2,4;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> 1</span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2">RAT</a> 2,4</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> 1</span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2">RAT</a> 2,4</span>)</span>, <font color="Maroon">B</font> =  <a href="borsuk_5.html#NK2" target="_self">].-infty,</a> 1, <font color="Maroon">C</font> =  <a href="borsuk_5.html#K2">RAT</a> 2,4 as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E2:77_2_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="borsuk_5.html#NK2" target="_self">].-infty,</a> 1
 
<b>by </b><i><a class="ref" href="isomichi.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:77_2_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="borsuk_5.html#NK1" target="_self">].-infty,</a> 1
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T77">BORSUK_5:77</a></i>;<br/>
<a NAME="E4:77_2_1"/><i><font color="Green">E43</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="borsuk_5.html#NK1" target="_self">].-infty,</a> 1
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T77">BORSUK_5:77</a></i>;<br/>
<a NAME="E5:77_2_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">2,4</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T54">BORSUK_5:54</a></i>;<br/>
<a NAME="E6:77_2_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a> 1</span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">2,4</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="isomichi.html#E4:77_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="pre_topc.html#T50">PRE_TOPC:50</a></i>;<br/>
<a NAME="E7:77_2_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> 1</span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">2,4</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="isomichi.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E8:77_2_1"/>
3 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">2,4</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E9:77_2_1"/><b>then </b>
3 <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="isomichi.html#E7:77_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E10:77_2_1"/><b>then </b><i><font color="Green">E45</font></i>: 
not  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="borsuk_5.html#T16">BORSUK_5:16</a></i>;<br/>
<a NAME="E11:77_2_1"/><i><font color="Green">E46</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="borsuk_5.html#T16">BORSUK_5:16</a></i>;<br/>
<a NAME="E12:77_2_1"/>
( not 1 <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#NK2" target="_self">].-infty,</a> 1 &amp; not 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">2,4</span><a href="rcomp_1.html#K2">.[</a></span> )
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T17">BORSUK_5:17</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E13:77_2_1"/><b>then </b>
not  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="isomichi.html#E11:77_2_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="isomichi.html#E7:77_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E14:77_2_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>, <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="isomichi.html#NR1" target="_self">are_c=-incomparable</a> 
 
<b>by </b><i><a class="txt" href="isomichi.html#E10:77_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<a NAME="E15:77_2_1"/><b>then </b>
<font color="Maroon">A</font> is <a href="isomichi.html#V5" target="_self">3rd_class</a>
 
<b>by </b><i><a class="ref" href="isomichi.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E16:77_2_1"/>
(  <a href="topmetr.html#K3">R^1</a>  is <a href="isomichi.html#V8" target="_self">with_3rd_class_subsets</a> &amp; not  <a href="topmetr.html#K3">R^1</a>  is <a href="struct_0.html#V3">empty</a> &amp;  <a href="topmetr.html#K3">R^1</a>  is <a href="pre_topc.html#V1">strict</a> )
 <b>by </b><i><a class="ref" href="isomichi.html#T17" target="_self">Th17</a></i>;<br/>


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