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

<b>let </b><font color="Maroon">Gi</font> be  <a href="finset_1.html#V1">finite</a> non <a href="realset1.html#V1">trivial</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">li</font>, <font color="Olive">ri</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">li</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">ri</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">ri</font> &amp; ( for <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">xi</font> holds <br/> not <font color="Olive">xi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">ri</font> ) );<br/>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">li</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">ri</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">li</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> &amp; <font color="Maroon">ri</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> &amp; <font color="Maroon">li</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ri</font> )
 ;<br/><div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">li</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">ri</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:29_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font>
 ;<br/><b>thus </b><a NAME="E2:29_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">ri</font></span><a href="domain_1.html#K7">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">li</font>
;<br/>

<b>take </b>
<font color="Maroon">ri</font>
;<br/>

<b>thus </b><a NAME="E1:29_1_1_1"/>
( <font color="Maroon">li</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">ri</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">ri</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">ri</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> &amp; ( for <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">xi</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">ri</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">xi</font> holds <br/> not <font color="Olive">xi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> ) )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:29_1"/>
( <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> or <font color="Maroon">ri</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">li</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E4:29_1"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">ri</font></span><a href="domain_1.html#K7">}</a></span>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E2:29"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a> non <a href="realset1.html#V1">trivial</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Gi</font> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">B</font> be  <a href="finset_1.html#V1">finite</a> non <a href="realset1.html#V1">trivial</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Gi</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>] )
 ;<br/>

<b>then consider </b><font color="Maroon">li</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">ri</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:29_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">li</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">ri</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> &amp; ( for <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">xi</font> holds <br/> not <font color="Olive">xi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> ) )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">li</font> or ( <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> ) or <font color="Maroon">ri</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> )
 </span><b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1"><b>suppose </b></a><a NAME="E1:29_2_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">li</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">li</font>
;<br/><b>take </b>
<font color="Maroon">ri</font>
;<br/><b>thus </b><a NAME="E2:29_2_1_1"/>
( <font color="Maroon">li</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">ri</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>let </b><font color="Maroon">xi</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E3:29_2_1_1"/>
<font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E4:29_2_1_1"/><b>then </b>
( <font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></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="E5:29_2_1_1"/>
(  not <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">xi</font> or  not <font color="Maroon">xi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_2"><b>suppose </b></a><a NAME="E1:29_2_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">li</font>
;<br/><b>take </b>
<font color="Maroon">x</font>
;<br/><a NAME="E2:29_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:29_2_1_2"/>
( <font color="Maroon">li</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>let </b><font color="Maroon">xi</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E4:29_2_1_2"/>
<font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E5:29_2_1_2"/><b>then </b>
( <font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:29_2_1_2"/><b>then </b>
(  not <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">xi</font> or  not <font color="Maroon">xi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> or <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:29_2_1_2"/>
(  not <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">xi</font> or  not <font color="Maroon">xi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_3"><b>suppose </b></a><a NAME="E1:29_2_1_3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">ri</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">li</font>
;<br/><b>take </b>
<font color="Maroon">ri</font>
;<br/><b>thus </b><a NAME="E2:29_2_1_3"/>
( <font color="Maroon">li</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">ri</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>let </b><font color="Maroon">xi</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E3:29_2_1_3"/>
<font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E4:29_2_1_3"/><b>then </b>
( <font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></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="E5:29_2_1_3"/>
(  not <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">xi</font> or  not <font color="Maroon">xi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E3:29"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Gi</font>]
 <b>from </b><i><a class="ref" href="chain_1.html#S3" target="_self">CHAIN_1:sch 3</a>(, );</i><br/>


</div>
