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

<b>let </b><font color="Maroon">S1</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">S2</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:87_1_2"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">a</font> being  <a href="supinf_1.html#NM3">R_eal</a> holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font> iff  ex <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Olive">A</font> ) )
 <b>and </b><br/><a NAME="E2:87_1_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">a</font> being  <a href="supinf_1.html#NM3">R_eal</a> holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font> iff  ex <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Olive">A</font> ) )
 ;<br/>

<a NAME="E3:87_1_2"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font> iff <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_2_1"><b>proof </b></a><div class="add">

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:87_1_2_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/><a NAME="E3:87_1_2_1_1"/>
 ex <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Olive">A</font> )
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D15" target="_self">Def15</a></i>;<br/><b>hence </b><a NAME="E4:87_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T55" target="_self">Th55</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:87_1_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E4:87_1_2_1"/>
 ex <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Olive">A</font> )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T55" target="_self">Th55</a>, <a class="ref" href="supinf_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>hence </b><a NAME="E5:87_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:87_1_2"/>
<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
