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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:95"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#K2" target="_self">RAT</a> 2,4</span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">4,5</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">5</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span>
 ;<br/>

<b>then </b><i><font color="Green">E77</font></i>: <font color="Maroon">c<sub>1</sub></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"><span class="p3">(<span class="default"><a href="borsuk_5.html#K2" target="_self">RAT</a> 2,4</span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default">4,5</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">5</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_5.html#T13" target="_self">Th13</a></i>
<br/>.= 
<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#K2" target="_self">RAT</a> 2,4</span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default">4,5</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">5</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span></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"><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" target="_self">RAT</a> 2,4</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">4,5</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">5</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span></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"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default">4</span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">4,5</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">5</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_5.html#T85" target="_self">Th85</a></i>
;<br/>
<a NAME="E3:95"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">4</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#R1">misses</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">4</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E4:95"/><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">4</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4 )
 </span><b>by </b><i><a class="txt" href="borsuk_5.html#E1:95_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_2_1_1"><b>suppose </b></a><a NAME="E1:95_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span>
 ;<br/><div class="add"><a NAME="E2:95_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 <b>by </b><i><a class="ref" href="borsuk_5.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E3:95_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 4
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:95_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">4</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_2_1_2"><b>suppose </b></a><a NAME="E1:95_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4
 ;<br/><div class="add"><a NAME="E2:95_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4
 <b>by </b><i><a class="ref" href="borsuk_5.html#T53" target="_self">Th53</a></i>;<br/><b>hence </b><a NAME="E3:95_2_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">4</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:95"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">4,5</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">5</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">4</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">4,5</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">5</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_3_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">4,5</span><a href="rcomp_1.html#K2">.[</a></span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">5</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span> )
 </span><b>by </b><i><a class="txt" href="borsuk_5.html#E1:95_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_3_1_1"><b>suppose </b></a><a NAME="E1:95_3_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">4,5</span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><div class="add"><a NAME="E2:95_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E3:95_3_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">4</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_3_1_2"><b>suppose </b></a><a NAME="E1:95_3_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">5</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span>
 ;<br/><div class="add"><a NAME="E2:95_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 5
 <b>by </b><i><a class="ref" href="borsuk_5.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E3:95_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:95_3_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">4</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:95"/><b>then </b>
<span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</span>)</span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">4,5</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">5</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:95"><i><font color="Green">E78</font></i></a>, <a class="txt" href="borsuk_5.html#E4:95"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_1.html#T64">XBOOLE_1:64</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  = 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default">4</span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default">4,5</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default">5</span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_5.html#E2:95"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T87">XBOOLE_1:87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</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">4</span><a href="seq_4.html#K1">}</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">5</span><a href="seq_4.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_5.html#T88" target="_self">Th88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">4</span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">5</span><a href="seq_4.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
;<br/>
<b>hence </b><a NAME="E8:95"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">4</span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">5</span><a href="seq_4.html#K1">}</a></span>
 ;<br/>


</div>
