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

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







<b>assume </b><a NAME="E1:50"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:50"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1"/>
for <font color="Olive">r</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> holds <br/><font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:50_1_1"/><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_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="E1:50_1_1_1_1"/>
( <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> or  not <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_1_1_1_1"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:50_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E3:50_1_1_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_1_1_1_2"/>
not <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:50_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:50_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:50_1_1_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:50_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_1"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:50"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">b</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E6:50"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<a NAME="E1:50_2"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_2_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> ) </span>}</span>  )
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:50_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:50_2_1"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:50_2"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:50_2_2"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> ) </span>}</span>  )
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:50_2_2"/><b>then </b>
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:50_2_2"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:50_2"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:50"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
