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<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> , <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="E1:80"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:80"/>
 <a href="supinf_1.html#K6" target="_self">sup</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> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<a NAME="E1:80_1"/><i><font color="Green">E23</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="E2:80_1"/>
 <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="E3:80_1"/><b>then </b>
 <a href="supinf_1.html#K6" target="_self">sup</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">S</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D13" target="_self">Def13</a>, <a class="ref" href="supinf_1.html#T51" target="_self">Th51</a>, <a class="ref" href="supinf_1.html#T47" target="_self">Th47</a></i>;<br/>
<b>hence </b><a NAME="E4:80_1"/>
 <a href="supinf_1.html#K6" target="_self">sup</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> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T33" target="_self">Th33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:80"/>
 <a href="supinf_1.html#K6" target="_self">sup</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> <font color="Maroon">S</font>
 ;<br/>


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