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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">R</font> be  <a href="waybel_4.html#V1" target="_self">auxiliary(i)</a> <a href="waybel_4.html#V2" target="_self">auxiliary(ii)</a> <a href="waybel_4.html#V3" target="_self">auxiliary(iii)</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:130"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> <a href="waybel_4.html#NR2" target="_self">satisfies_INT</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D21" target="_self">WAYBEL_4:def 21</a><br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E2:130"/><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</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">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 ;<br/>

<b>then consider </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><br/><a NAME="E4:130"/><i><font color="Green">E33</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <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">R</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </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><br/><a NAME="E6:130"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font> &amp; <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">R</font> &amp; <font color="Maroon">x</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#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E7:130"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">y'</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<b>take </b><font color="Maroon">Y</font> = <font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y'</font>;<br/>

<a NAME="E8:130"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E9:130"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="130_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:130_1"/>
( <font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> or  not <font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="130_1_1"><b>suppose </b></a><a NAME="E1:130_1_1"/>
<font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:130_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T32">ORDERS_2:32</a></i>;<br/><b>hence </b><a NAME="E3:130_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</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#T12" target="_self">Th12</a>, , , , , <a class="ref" href="orders_2.html#T32">ORDERS_2:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="130_1_2"><b>suppose </b></a><a NAME="E1:130_1_2"/>
not <font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:130_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>;<br/><a NAME="E3:130_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><b>hence </b><a NAME="E4:130_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</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#T12" target="_self">Th12</a>, , , , , , <a class="ref" href="orders_2.html#T32">ORDERS_2:32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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