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<div class="add">

<b>let </b><font color="Maroon">F</font> be    <a href="supinf_1.html#M3" target="_self">bool_DOMAIN</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">S</font> be  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> ;<br/>



<b>assume </b><a NAME="E1:84"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:84"/>
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K8" target="_self">SUP</a> <font color="Maroon">F</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:84_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K8" target="_self">SUP</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">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> <b> such that </b><br/><a NAME="E3:84_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="supinf_1.html#T51" target="_self">Th51</a></i>;<br/>
<a NAME="E4:84_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:84_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>hence </b><a NAME="E2:84_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:84_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D10" target="_self">Def10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:84"/>
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">S</font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of  <a href="supinf_1.html#K8" target="_self">SUP</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


</div>
