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



<b>assume </b><a NAME="E1:34"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1">Real</a>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">xi</font>;<br/>
<b>set </b><font color="Maroon">Gi'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">ri'</font>) where <font color="Olive">ri'</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">ri'</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">ri'</font>] ) </span>}</span> ;<br/>
<a NAME="E2:34"/><i><font color="Green">E51</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">ri'</font>) where <font color="Olive">ri'</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">ri'</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">ri'</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Gi</font>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S17">FRAENKEL:sch 17</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">Gi'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">ri'</font>) where <font color="Olive">ri'</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">ri'</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">ri'</font>] ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1"/>
( <font color="Maroon">Gi'</font> is <a href="xboole_0.html#V1">zero</a> or  not <font color="Maroon">Gi'</font> is <a href="xboole_0.html#V1">zero</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>suppose </b></a><a NAME="E1:34_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">Gi'</font> is <a href="xboole_0.html#V1">zero</a>
 ;<br/><div class="add"><div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xi'</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:34_1_1_1"/>
( <font color="Maroon">xi'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> &amp; <font color="Maroon">xi'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">xi</font> )
 ;<br/><a NAME="E2:34_1_1_1"/><b>then </b>
<font color="Maroon">xi'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi'</font>
 ;<br/><b>hence </b><a NAME="E3:34_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">li</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:34_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">li</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</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/><font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">xi'</font> ) )
 <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">li</font>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><b>now </b></a><div class="add"><div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_1_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">li</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi</font>
 ;<br/><a NAME="E2:34_1_1_2_1"/>
for <font color="Olive">xi'</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">xi'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> iff <font color="Olive">xi'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_1_1"><b>proof </b></a><div class="add">

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1_1_2_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">xi'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font>
 ;<br/><b>then reconsider </b><font color="Maroon">xi''</font> = <font color="Maroon">xi'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E3:34_1_1_2_1_1_1"/>
( <font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xi''</font> &amp; <font color="Maroon">xi''</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xi</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:34_1_1_2_1_1_1"/>
<font color="Maroon">xi'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:34_1_1_2_1_1"/>
( <font color="Maroon">xi'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi</font> implies <font color="Maroon">xi'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:34_1_1_2_1"/>
<font color="Maroon">Gi</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">xi</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="E4:34_1_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E2:34_1_1_2"/>
<font color="Maroon">li</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xi</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:34_1_1_2"/>
( <font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> &amp; <font color="Maroon">li</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="Maroon">xi</font> )
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>thus </b><a NAME="E4:34_1_1_2"/>
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">xi</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">li</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:34_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">xi</font>,<font color="Maroon">li</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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>suppose </b></a><a NAME="E1:34_1_2"/>
not <font color="Maroon">Gi'</font> is <a href="xboole_0.html#V1">zero</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">Gi'</font> = <font color="Maroon">Gi'</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/><b>consider </b><font color="Maroon">ri</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:34_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">ri</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi'</font> &amp; ( for <font color="Olive">ri'</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">ri'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi'</font> holds <br/><font color="Olive">ri'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">ri</font> ) )
 <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">ri</font>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:34_1_2_1"/>
<font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E2:34_1_2_1"/>
<font color="Maroon">ri</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:34_1_2_1"/>
 ex <font color="Olive">ri'</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">ri'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ri</font> &amp; <font color="Olive">ri'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font> &amp; <font color="Maroon">xi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">ri'</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:34_1_2_1"/>
<font color="Maroon">xi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ri</font>
 ;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">xi'</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:34_1_2_1_1"/>
<font color="Maroon">xi'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi</font>
 ;<br/><a NAME="E2:34_1_2_1_1"/><b>then </b>
( <font color="Maroon">xi'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xi</font> or <font color="Maroon">xi'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Gi'</font> )
 ;<br/><b>hence </b><a NAME="E3:34_1_2_1_1"/>
(  not <font color="Maroon">xi</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/>;<br/></div>
<b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:34_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">xi</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><b>end;</b></div></div><b>end;</b></div>

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