<?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">z</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>



<a NAME="E1:32"/>
(  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">X</font> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</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#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:32_1"/>
(  <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">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  <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> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
;<br/>
<b>hence </b><a NAME="E5:32_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">z</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="ref" href="supinf_2.html#D3" target="_self">Def3</a></i>;<br/>


</div>
