<?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>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>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="E1:33"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">Gi</font> <a href="hidden.html#R1">=</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>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">li'</font>,<font color="Maroon">ri'</font></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">Gi</font>
 ;<br/><a NAME="E2:33_1"/><b>then </b>
( <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> )
 <b>by </b><i/>;<br/><a NAME="E3:33_1"/><b>then </b>
( ( <font color="Maroon">li'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">li</font> or <font color="Maroon">li'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ri</font> ) &amp; ( <font color="Maroon">ri'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">li</font> or <font color="Maroon">ri'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ri</font> ) &amp; <font color="Maroon">li'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ri'</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:33_1"/>
( ( <font color="Maroon">li'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">li</font> &amp; <font color="Maroon">ri'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ri</font> ) or ( <font color="Maroon">li'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ri</font> &amp; <font color="Maroon">ri'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">li</font> ) )
 ;<br/></div>
<b>end;</b></div>

<a NAME="E3:33"/>
for <font color="Olive">Gi</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> <br/> for <font color="Olive">li</font>, <font color="Olive">ri</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">Gi</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">li</font>,<font color="Olive">ri</font></span><a href="domain_1.html#K7">}</a></span> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">li</font>,<font color="Olive">ri</font></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Olive">Gi</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><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>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="E1:33_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">Gi</font> <a href="hidden.html#R1">=</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>
 ;<br/>

<b>take </b>
<font color="Maroon">li</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="chain_1.html#D6" target="_self">CHAIN_1:def 6</a><br/>

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

<b>thus </b><a NAME="E2:33_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">ri</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">li</font>,<font color="Maroon">ri</font></span><a href="domain_1.html#K1">]</a></span> &amp; <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> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>

<a NAME="E3:33_2"/>
<font color="Maroon">li</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ri</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:33_2"/>
( ( <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">Gi</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> ) ) or ( <font color="Maroon">ri</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">li</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">Gi</font> holds <br/>( not <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">xi</font> &amp; 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="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33"/>
( ( ( <font color="Maroon">li'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">li</font> &amp; <font color="Maroon">ri'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ri</font> ) or ( <font color="Maroon">li'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ri</font> &amp; <font color="Maroon">ri'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">li</font> ) ) implies <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">li'</font>,<font color="Maroon">ri'</font></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">Gi</font> )
 <b>by </b><i/>;<br/>


</div>
