<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">L</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">L</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
 ;<br/>

<a NAME="E3:15"/><i><font color="Green">E37</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:15"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:15"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>


</div>
