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

<b>assume </b><a NAME="E1:44"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> </span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>set </b><font color="Maroon">UB</font> =  <a href="supinf_1.html#K2" target="_self">-infty</a> ;<br/>
<a NAME="E2:44"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">a</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#T22" target="_self">Th22</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>take </b>
 <a href="supinf_1.html#K2" target="_self">-infty</a> 
;<br/>

<b>thus </b><a NAME="E3:44"/>
(  <a href="supinf_1.html#K2" target="_self">-infty</a>  is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">X</font> &amp; ( for <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">y</font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">X</font> holds <br/> <a href="supinf_1.html#K2" target="_self">-infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font> ) )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T5">XXREAL_0:5</a></i>;<br/>


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