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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</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>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">C</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green">E30</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">C</font> is <a href="waybel35.html#V2" target="_self">maximal</a>
 ;<br/>

<b>set </b><font color="Maroon">s</font> =  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1"/>
(  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or  not  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1"><b>suppose </b></a><a NAME="E1:33_1_1"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2"><b>suppose </b></a><a NAME="E1:33_1_2"/>
not  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:33_1_2"/><b>then </b>
 ex <font color="Olive">cs</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">cs</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span>,<font color="Olive">cs</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp;  ex <font color="Olive">cs1</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">C</font></span>)</span> st <br/>( <font color="Olive">cs</font> <a href="hidden.html#R1">=</a> <font color="Olive">cs1</font> &amp; <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">cs1</font> &amp; ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">C</font></span>)</span>  st <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">a</font> holds <br/><font color="Olive">cs1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> ) ) )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:33_1_2"/>
 <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><a class="ref" href="yellow_0.html#T15">YELLOW_0:15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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