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<b>let </b><font color="Maroon" title="c1">A</font> be   <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> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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>assume </b><a NAME="E1:66"/><i><font color="Green" title="E34">A1</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK1" title="BORSUK_5:NK.1">].-infty,</a> <font color="Maroon" title="c2">a</font></span>)</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"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">c</font> )
 ;<br/>

<b>reconsider </b><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"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></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#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c4">c</font> 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>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E3:66"/><i><font color="Green" title="E35">A2</font></i>: 
 <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#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="isomichi.html#E1:66"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="borsuk_5.html#T36" title="BORSUK_5:th.36">BORSUK_5:36</a></i>;<br/>
<a NAME="E4:66"/><i><font color="Green" title="E36">A3</font></i>: 
 <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>  <a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T75" title="BORSUK_5:th.75">BORSUK_5:75</a></i>;<br/>
<a NAME="E5:66"/>
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="isomichi.html#E1:66"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:66"/><b>then </b>
 <a href="borsuk_5.html#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c4">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="borsuk_5.html#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="limfunc1.html#T8" title="LIMFUNC1:th.8">LIMFUNC1:8</a></i>;<br/>
<a NAME="E7:66"/><b>then </b><i><font color="Green" title="E37">A4</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_5.html#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="borsuk_5.html#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  = 
<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#NK1" title="BORSUK_5:NK.1">].-infty,</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="isomichi.html#E1:66"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="borsuk_5.html#T83" title="BORSUK_5:th.83">BORSUK_5:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#NK1" title="BORSUK_5:NK.1">].-infty,</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_5.html#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="limfunc1.html#T24" title="LIMFUNC1:th.24">LIMFUNC1:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_5.html#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="isomichi.html#T40" target="_self" title="ISOMICHI:th.40">Th40</a>, <a class="txt" href="isomichi.html#E1:66"><i><font color="Green" title="E34">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T52" title="XBOOLE_1:th.52">XBOOLE_1:52</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></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#NK4" title="BORSUK_5:NK.4">].</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="txt" href="isomichi.html#E7:66"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="isomichi.html#T42" target="_self" title="ISOMICHI:th.42">Th42</a></i>
;<br/>
<a NAME="E9:66"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <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"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</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#NK3" title="BORSUK_5:NK.3">[.</a> <font color="Maroon" title="c4">c</font></span>)</span>
 
<b>by </b><i><a class="txt" href="isomichi.html#E3:66"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="isomichi.html#E4:66"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="pre_topc.html#T50" title="PRE_TOPC:th.50">PRE_TOPC:50</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  = 
<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_5.html#T83" title="BORSUK_5:th.83">BORSUK_5:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" title="BORSUK_5:NK.2">].-infty,</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="limfunc1.html#T24" title="LIMFUNC1:th.24">LIMFUNC1:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" title="BORSUK_5:NK.2">].-infty,</a> <font color="Maroon" title="c2">a</font></span>)</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"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<b>by </b><i><a class="txt" href="isomichi.html#E1:66"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="isomichi.html#T43" target="_self" title="ISOMICHI:th.43">Th43</a></i>
;<br/>
<b>hence </b><a NAME="E11:66"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" title="BORSUK_5:NK.2">].-infty,</a> <font color="Maroon" title="c2">a</font></span>)</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"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="tops_1.html#D1" title="TOPS_1:def.1">TOPS_1:def 1</a></i>;<br/>


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