<?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="56_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_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>1</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="56_1_1"><b>suppose </b></a><a NAME="E1:56_1_1"/><i><font color="Green">E25</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>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="E2:56_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<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="E4:56_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="rat_1.html#V1">irrational</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</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> )
 <b>by </b><i><a class="txt" href="isomichi.html#E2:56_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="borsuk_5.html#T53">BORSUK_5:53</a>, <a class="txt" href="isomichi.html#E1:56_1_1"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E5:56_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T15">BORSUK_5:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2"><b>suppose </b></a><a NAME="E1:56_1_2"/>
<font color="Maroon">c<sub>1</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="E2:56_1_2"/><b>then </b>
 <a href="borsuk_5.html#K3">IRRAT</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="isomichi.html#T35" target="_self">Th35</a></i>;<br/><b>hence </b><a NAME="E3:56_1_2"/>
 <a href="borsuk_5.html#K3">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>  <a href="borsuk_5.html#NK3" target="_self">[.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
