<?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> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">a</font> is    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">X</font>
 ;<br/><a NAME="E2:39_1"/>
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">a</font> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_1"/>
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">X</font> holds <br/> <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:39_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K6" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:39"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E3:39_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:39_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<a NAME="E5:39_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:39_1_1_1"/>
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_1_1"/>
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">a</font> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:39_1"/>
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">a</font> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">X</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:39"/><i><font color="Green">E28</font></i>: 
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">a</font> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:39"/>
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:39_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:39_2"/><b>then </b>
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:39"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E3:39_2"/><b>then </b>
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T3" target="_self">Th3</a>, <a class="ref" href="supinf_1.html#D10">SUPINF_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E4:39_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
<font color="Maroon">a</font> is    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>


</div>
