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

<b>assume </b><a NAME="E1:26"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 ;<br/>

<b>assume </b><a NAME="E2:26"/>
<font color="Maroon">Y</font> is <a href="supinf_1.html#V4" target="_self">bounded_below</a>
 ;<br/>

<a NAME="E3:26"/><b>then </b>
 ex <font color="Olive">LB</font> being   <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">Y</font> st <font color="Olive">LB</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">LB</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E5:26"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">LB</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">LB</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  )
 ;<br/>
<a NAME="E6:26"/>
( <font color="Maroon">LB</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">LB</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T11" target="_self">Th11</a>, , </i>;<br/>
<b>hence </b><a NAME="E7:26"/>
<font color="Maroon">X</font> is <a href="supinf_1.html#V4" target="_self">bounded_below</a>
 <b>by </b><i><a class="ref" href="supinf_1.html#D4" target="_self">Def4</a></i>;<br/>


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