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<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">ri'</font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<a NAME="E1:36"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">ri</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ri'</font> &amp; <font color="Maroon">ri'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ri</font> implies <font color="Maroon">ri</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ri'</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>assume </b><a NAME="E2:36"/><i><font color="Green">E51</font></i>: 
( <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> &amp; <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="E3:36"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">ri</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/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1"/>
( ( <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> ) ) ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><b>suppose </b></a><a NAME="E1:36_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> &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> ) )
 ;<br/><div class="add"><a NAME="E2:36_1_1"/>
( <font color="Maroon">ri'</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">ri'</font> &amp; <font color="Maroon">ri'</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">ri'</font> )
 ;<br/><b>hence </b><a NAME="E3:36_1_1"/>
<font color="Maroon">ri</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ri'</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2"><b>suppose </b></a><a NAME="E1:36_1_2"/><i><font color="Green">E54</font></i>: 
( <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> ) ) )
 ;<br/><div class="add"><a NAME="E2:36_1_2"/>
( <font color="Maroon">ri'</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">ri'</font> &amp; <font color="Maroon">ri'</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">ri'</font> )
 ;<br/><b>hence </b><a NAME="E3:36_1_2"/>
<font color="Maroon">ri</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ri'</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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