<?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/>

<a NAME="E1:31"/>
for <font color="Olive">z</font> being  <a href="supinf_1.html#NM3">R_eal</a> holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff  ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <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> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Olive">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:31_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies  ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <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> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Olive">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>set </b><font color="Maroon">x</font> =  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">z</font>;<br/>
<a NAME="E2:31_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</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">z</font></span>)</span>
 
;<br/>
<a NAME="E3:31_1_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>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:31_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E4:31_1_1"/>
 ex <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <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> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Olive">x</font> )
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:31_1"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">x</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><a NAME="E2:31_1"/><i><font color="Green">E29</font></i>: 
( <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> &amp; <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> )
 ;<br/>

<a NAME="E3:31_1"/>
 ex <font color="Olive">a</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Olive">a</font> )
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E4:31_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="E2:31"/>
 <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> <a href="hidden.html#R1">=</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>
