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<span class="kw">set </span><font color="Maroon" title="c1">B</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">C</font> =  <a href="borsuk_5.html#K2" title="BORSUK_5:func.2">RAT</a> 2,4;<br/>
<span class="kw">take </span>
 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  is  <a href="isomichi.html#V8" title="ISOMICHI:attr.8">with_3rd_class_subsets</a>  &amp; not  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  is  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">A</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2" title="BORSUK_5:func.2">RAT</a> 2,4</span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">A</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2" title="BORSUK_5:func.2">RAT</a> 2,4</span>)</span>, <font color="Maroon" title="c5">B</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>, <font color="Maroon" title="c6">C</font> =  <a href="borsuk_5.html#K2" title="BORSUK_5:func.2">RAT</a> 2,4 as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<a NAME="E2:77_2_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">2,4</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_5.html#T54" title="BORSUK_5:th.54">BORSUK_5:54</a></span>;<br/>
<a NAME="E3:77_2_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_5.html#T77" title="BORSUK_5:th.77">BORSUK_5:77</a></span>;<br/>
<a NAME="E4:77_2_1"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">2,4</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77_2_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="pre_topc.html#T50" title="PRE_TOPC:th.50">PRE_TOPC:50</a></span>;<br/>
<a NAME="E5:77_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A3</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">2,4</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="isomichi.html#T44" target="_self" title="ISOMICHI:th.44">Th44</a></span>;<br/>
<a NAME="E6:77_2_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,1</span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="isomichi.html#T37" target="_self" title="ISOMICHI:th.37">Th37</a>, <a class="ref" href="borsuk_5.html#T77" title="BORSUK_5:th.77">BORSUK_5:77</a></span>;<br/>
<a NAME="E7:77_2_1"/>
3 <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">2,4</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E8:77_2_1"/><span class="kw">then </span>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77_2_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E9:77_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A5</font></span>: 
not  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:77_2_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></span>;<br/>
<a NAME="E10:77_2_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
not 1 <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">2,4</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E11:77_2_1"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
not 1 <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"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/>
<a NAME="E12:77_2_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:77_2_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></span>;<br/>
<a NAME="E13:77_2_1"/><span class="kw">then </span>
not  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">A</font></span>)</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> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77_2_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E11:77_2_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E10:77_2_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E14:77_2_1"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">A</font></span>)</span>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="isomichi.html#NR1" title="ISOMICHI:NR.1">are_c=-incomparable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:77_2_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></span>;<br/>
<a NAME="E15:77_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">A</font> is  <a href="isomichi.html#V5" title="ISOMICHI:attr.5">3rd_class</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="isomichi.html#D8" target="_self" title="ISOMICHI:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:77_2_1"/>
(  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  is  <a href="isomichi.html#V8" title="ISOMICHI:attr.8">with_3rd_class_subsets</a>  &amp; not  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  is  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="isomichi.html#D11" target="_self" title="ISOMICHI:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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