<?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="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">R</font> be  <a href="waybel_4.html#V1">auxiliary(i)</a> <a href="waybel_4.html#V2">auxiliary(ii)</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="waybel35.html#M1" target="_self">strict_chain</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">C</font> is <a href="waybel35.html#V2" target="_self">maximal</a>
 ;<br/>

<a NAME="E2:35"/>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">C</font></span>)</span> holds   <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">X</font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">C</font></span>)</span>;<br/>

<a NAME="E1:35_1"/>
( <font color="Maroon">X</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">C</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>hence </b><a NAME="E2:35_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">C</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:35"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">C</font> is <a href="lattice3.html#V3">complete</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#T53">YELLOW_0:53</a></i>;<br/>


</div>
