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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">AR</font> be   <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:126"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">AR</font> <a href="waybel_4.html#NR1" target="_self">satisfies_SI</a> 
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>given </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E2:126"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">y</font> <a href="waybel_4.html#R2" target="_self">is_maximal_wrt</a> <font color="Maroon">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font>,<font color="Maroon">AR</font>
 ;<br/>

<a NAME="E3:126"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">y'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  holds <br/>(  not <font color="Olive">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font> or  not <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y'</font> or  not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Olive">y'</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> ) ) )
 
<b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E4:126"/><i><font color="Green">E34</font></i>: 
not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font>
 ;<br/>

<a NAME="E5:126"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="126_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:126_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="126_1_1"><b>suppose </b></a><a NAME="E1:126_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:126_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="waybel_4.html#T12" target="_self">Th12</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="126_1_2"><b>suppose </b></a><a NAME="E1:126_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:126_1_2"/><i><font color="Green">E40</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_4.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E4:126_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:126_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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