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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> , <font color="Maroon">B</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="E1:8"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font>
 ;<br/>

<b>assume </b><a NAME="E2:8"/>
not  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">s1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E4:8"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">B</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s1</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s2</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E6:8"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s1</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E7:8"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


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