<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>, <font color="Maroon">C</font> =  <a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>, <font color="Maroon">D</font> =  <a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">c</font> 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="E2:57"/>
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T79">TOPREAL6:79</a></i>;<br/>
<a NAME="E3:57"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E4:57"/>
 <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">c</font> is <a href="rcomp_1.html#V2">closed</a>
 
<b>by </b><i><a class="ref" href="fcont_3.html#T5">FCONT_3:5</a></i>;<br/>
<a NAME="E5:57"/><b>then </b>
<font color="Maroon">D</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T79">TOPREAL6:79</a></i>;<br/>
<a NAME="E6:57"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>assume </b><a NAME="E7:57"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2">RAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> )
 ;<br/>

<b>then </b><i><font color="Green">E44</font></i>: <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  = 
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#K2">RAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_5.html#T83">BORSUK_5:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#K2">RAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T41">XBOOLE_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a> <font color="Maroon">b</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_5.html#T85">BORSUK_5:85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a> <font color="Maroon">b</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a> <font color="Maroon">b</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>
;<br/>
<i><font color="Green">E45</font></i>: <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="isomichi.html#D3" target="_self">Def3</a>, </i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>
;<br/>
<a NAME="E10:57"/><i><font color="Green">E46</font></i>: 
 <a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="xboole_0.html#R1">misses</a>  <a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>proof </b></a><div class="add">

<a NAME="E1:57_3"/><i><font color="Green">E47</font></i>: 
 <a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="tarski.html#R1">c=</a>  <a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="isomichi.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E2:57_3"/>
 <a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">b</font> <a href="xboole_0.html#R1">misses</a>  <a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="isomichi.html#T4" target="_self">Th4</a>, <a class="ref" href="isomichi.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E3:57_3"/>
 <a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="xboole_0.html#R1">misses</a>  <a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="isomichi.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:57"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a>, <a class="txt" href="isomichi.html#E6:57"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E12:57"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="isomichi.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:57"/><b>then </b>
 <a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">c</font> <a href="xboole_0.html#R1">misses</a>  <a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="isomichi.html#T1" target="_self">Th1</a></i>;<br/>
<b>then </b><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="isomichi.html#T5" target="_self">Th5</a>, <a class="txt" href="isomichi.html#E3:57"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c</font></span><a href="seq_4.html#K1">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="borsuk_5.html#NK4" target="_self">].</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_5.html#T67">BORSUK_5:67</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c</font></span><a href="seq_4.html#K1">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#NK4" target="_self">].</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_5.html#T67">BORSUK_5:67</a></i>
;<br/>
<b>then </b> <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T50">PRE_TOPC:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T50">PRE_TOPC:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="isomichi.html#D2" target="_self">Def2</a>, <a class="ref" href="isomichi.html#T3" target="_self">Th3</a>, <a class="ref" href="isomichi.html#T4" target="_self">Th4</a>, <a class="ref" href="borsuk_5.html#T55">BORSUK_5:55</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="isomichi.html#T4" target="_self">Th4</a>, <a class="ref" href="heine.html#T2">HEINE:2</a></i>
<br/>.= 
 <a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="isomichi.html#T8" target="_self">Th8</a>, <a class="ref" href="borsuk_5.html#T28">BORSUK_5:28</a></i>
;<br/>
<a NAME="E16:57"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="borsuk_5.html#NK2" target="_self">].-infty,</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T99">BORSUK_5:99</a></i>;<br/>
<b>hence </b><a NAME="E17:57"/>
 <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> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="tops_1.html#D1">TOPS_1:def 1</a></i>;<br/>


</div>
