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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1"><b>suppose </b></a><a NAME="E1:91_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:91_1_1"/>
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</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="E1:91_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:91_1_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</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="91_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_1_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>1</sub></font></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/><a NAME="E3:91_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:91_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>1</sub></font></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/><a NAME="E3:91_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:91_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_3"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K3" target="_self">IRRAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_1_1_3_1_1"/>
(  not <font color="Maroon">c<sub>4</sub></font> is <a href="rat_1.html#V1">irrational</a> or <font color="Maroon">c<sub>4</sub></font> is <a href="rat_1.html#V1">irrational</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_3_1_1_1"/>
not <font color="Maroon">c<sub>4</sub></font> is <a href="rat_1.html#V1">irrational</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:91_1_1_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K3" target="_self">IRRAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E1:91_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="borsuk_5.html#E2:91_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="borsuk_5.html#E1:91_1_1_1_1_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="borsuk_5.html#T52" target="_self">Th52</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="rat_1.html#V1">irrational</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:91_1_1_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K3" target="_self">IRRAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E1:91_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="borsuk_5.html#E1:91_1_1_1_1_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="borsuk_5.html#T53" target="_self">Th53</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:91_1_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:91_1_1_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E3:91_1_1_1_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>let </b><font color="Maroon">c<sub>3</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="E3:91_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</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/><a NAME="E5:91_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span> )
 <b>by </b><i><a class="txt" href="borsuk_5.html#E3:91_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_2"/>
( <font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K3" target="_self">IRRAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span> )
 </span><b>by </b><i><a class="txt" href="borsuk_5.html#E5:91_1_1"><i><font color="Green">E76</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="91_1_1_2_1"><b>suppose </b></a><a NAME="E1:91_1_1_2_1"/>
<font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_5.html#NK1" target="_self">.]</a></span>
 ;<br/><div class="add"><a NAME="E2:91_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E3:91_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_5.html#E1:91_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="borsuk_5.html#T52" target="_self">Th52</a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>hence </b><a NAME="E4:91_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2_2"><b>suppose </b></a><a NAME="E1:91_1_1_2_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K3" target="_self">IRRAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:91_1_1_2_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E3:91_1_1_2_2"/>
 <a href="borsuk_5.html#K3" target="_self">IRRAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E1:91_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="borsuk_5.html#T84" target="_self">Th84</a></i>;<br/><a NAME="E4:91_1_1_2_2"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E1:91_1_1_2_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E5:91_1_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E2:91_1_1_2_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2_3"><b>suppose </b></a><a NAME="E1:91_1_1_2_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 ;<br/><div class="add"><a NAME="E2:91_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E3:91_1_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_5.html#E1:91_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="borsuk_5.html#T52" target="_self">Th52</a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>hence </b><a NAME="E4:91_1_1_2_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2"><b>suppose </b></a><a NAME="E1:91_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:91_1_2"/><b>then </b>
<span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="borsuk_5.html#E105"><i><font color="Green">Lemma72</font></i></a></i>;<br/><a NAME="E3:91_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="borsuk_5.html#K3" target="_self">IRRAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 ;<br/><a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="borsuk_5.html#E104"><i><font color="Green">Lemma71</font></i></a>, <a class="txt" href="borsuk_5.html#E1:91_1_2"><i><font color="Green">E74</font></i></a></i>
<br/>.= 
 <a href="numbers.html#K1">REAL</a> 

<br/>.= 
<span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
<b>by </b><i><a class="txt" href="borsuk_5.html#E105"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="borsuk_5.html#E1:91_1_2"><i><font color="Green">E74</font></i></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"><font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:91_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</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"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</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="E5:91_1_2"/>
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="borsuk_5.html#K2" target="_self">RAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
