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<div class="add">

<b>let </b><font color="Maroon">X</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/>

<a NAME="E1:51"/>
( not <font color="Maroon">X</font> is <a href="supinf_1.html#V4" target="_self">bounded_below</a> implies  ex <font color="Olive">LB</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">LB</font> is    <a href="supinf_1.html#M2" target="_self">minorant</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#M2" target="_self">minorant</a> of <font color="Maroon">X</font> holds <br/><font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">LB</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">X</font> is <a href="supinf_1.html#V4" target="_self">bounded_below</a>
 ;<br/>

<b>take </b>
 <a href="supinf_1.html#K2" target="_self">-infty</a> 
;<br/>

<b>thus </b><a NAME="E2:51_1"/>
(  <a href="supinf_1.html#K2" target="_self">-infty</a>  is    <a href="supinf_1.html#M2" target="_self">minorant</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#M2" target="_self">minorant</a> of <font color="Maroon">X</font> holds <br/><font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  ) )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:51"/>
 ex <font color="Olive">LB</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">LB</font> is    <a href="supinf_1.html#M2" target="_self">minorant</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#M2" target="_self">minorant</a> of <font color="Maroon">X</font> holds <br/><font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">LB</font> ) )
 <b>by </b><i/>;<br/>


</div>
