<?xml version="1.0"?>
<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> , <font color="Maroon">Y</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>set </b><font color="Maroon">a</font> =  <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span>;<br/>
<a NAME="E1:78"/><i><font color="Green">E22</font></i>: 
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E2:78"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">Y</font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E3:78"/><b>then </b>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">Y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D13" target="_self">Def13</a>, </i>;<br/>
<a NAME="E4:78"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">Y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E5:78"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:78"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E7:78"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:78"/><b>then </b>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E9:78"/><b>then </b>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">Y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D14" target="_self">Def14</a>, </i>;<br/>
<b>hence </b><a NAME="E10:78"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">X</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">Y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="supinf_1.html#T51" target="_self">Th51</a>, </i>;<br/>


</div>
