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



<b>assume </b><a NAME="E1:6"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">A</font>
 ;<br/>

<b>assume </b><a NAME="E2:6"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">s</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/>not <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font>
 ;<br/>

<a NAME="E3:6"/>
<font color="Maroon">r</font> is    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_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>; <i><font color="Red">:: according to </font></i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a><br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>hence </b><a NAME="E2:6_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6"/>
contradiction
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>


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