<?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="E1:51"/><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="E2:51"/><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="E3:51"/><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">c</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#K9">/\</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="51_1"><b>proof </b></a><div class="add">

<a NAME="E1:51_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:51_1_1"/>
<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="E2:51_1_1"/><b>then </b>
<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">b</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:51_1_1"/><b>then </b>
( <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">b</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">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:51_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</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>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:51_1"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</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">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="51_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:51_1_2"/>
<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="E2:51_1_2"/><b>then </b>
<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">c</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/>;<br/>
<a NAME="E3:51_1_2"/><b>then </b>
( <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">c</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">c</font>,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:51_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</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">c</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="E3:51_1"/>
<span class="p1"><a href="rcomp_1.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c</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#K9">/\</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#T19">XBOOLE_1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:51"/>
<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#K9">/\</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">c</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<a NAME="E1:51_2"/>
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">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K9">/\</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> 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">c</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_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="E1:51_2_1"/><i><font color="Green">E51</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">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> <a href="subset_1.html#K9">/\</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/>

<a NAME="E2:51_2_1"/><b>then </b><i><font color="Green">E52</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">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span> &amp; <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">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_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E4:51_2_1"/>
(  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">b</font> ) &amp;  ex <font color="Olive">q</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">c</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> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E5:51_2_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E6:51_2_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">c</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1" target="_self">.]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:51_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#K9">/\</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">c</font>,<font color="Maroon">b</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>
<b>hence </b><a NAME="E5:51"/>
<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#K9">/\</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">c</font>,<font color="Maroon">b</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>
