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<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">R</font> be  <a href="waybel_4.html#V1" title="WAYBEL_4:attr.1">auxiliary(i)</a> <a href="waybel_4.html#V2" title="WAYBEL_4:attr.2">auxiliary(ii)</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">C</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font>;<br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="waybel35.html#V2" title="WAYBEL35:attr.2">maximal</a>
 ;<br/>

<a NAME="E2:35"/>
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span> holds   <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">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" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>;<br/>

<a NAME="E1:35_1"/>
( <font color="Maroon" title="c4">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<b>hence </b><a NAME="E2:35_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>, <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E1:35"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="waybel35.html#T11" target="_self" title="WAYBEL35:th.11">Th11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:35"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#T53" title="YELLOW_0:th.53">YELLOW_0:53</a></i>;<br/>


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