<?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#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="yellow_0.html#V8">sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:5"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:5"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">V</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">V</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">V</font>,<font color="Maroon">L</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/>
<b>then consider </b><font color="Maroon">Z</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:5"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E6:5"/><i><font color="Green">E35</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Z1</font> = <font color="Maroon">Z</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:5"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z1</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D19">YELLOW_0:def 19</a></i>;<br/>
<a NAME="E10:5"/><b>then </b>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Z1</font>,<font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z1</font>,<font color="Maroon">S</font> )
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T65">YELLOW_0:65</a></i>;<br/>
<a NAME="E11:5"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">V</font>,<font color="Maroon">S</font></span>)</span> where <font color="Olive">V</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">V</font>,<font color="Maroon">S</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E12:5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/>


</div>
