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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">AR</font> be  <a href="waybel_4.html#V2" title="WAYBEL_4:attr.2">auxiliary(ii)</a> <a href="waybel_4.html#V3" title="WAYBEL_4:attr.3">auxiliary(iii)</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:128"/><i><font color="Green" title="E76">A1</font></i>: 
<font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#NR2" title="WAYBEL_4:NR.2">satisfies_INT</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D23" target="_self" title="WAYBEL_4:def.23">WAYBEL_4:def 23</a><br/><b>let </b><font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><a NAME="E2:128"/>
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font> )
 ;<br/>

<a NAME="E3:128"/><b>then </b><i><font color="Green" title="E77">A2</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font> )
 
<b>by </b><i><a class="ref" href="waybel_4.html#T13" target="_self" title="WAYBEL_4:th.13">Th13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">y'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:128"/><i><font color="Green" title="E78">A3</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c6">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">y'</font>,<font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font> )
 <b>by </b><i><a class="txt" href="waybel_4.html#E1:128"><i><font color="Green" title="E76">A1</font></i></a>, <a class="ref" href="waybel_4.html#D22" target="_self" title="WAYBEL_4:def.22">Def22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">z'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E7:128"/><i><font color="Green" title="E79">A4</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c7">z'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">z'</font>,<font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font> )
 <b>by </b><i><a class="txt" href="waybel_4.html#E1:128"><i><font color="Green" title="E76">A1</font></i></a>, <a class="txt" href="waybel_4.html#E3:128"><i><font color="Green" title="E77">A2</font></i></a>, <a class="ref" href="waybel_4.html#D22" target="_self" title="WAYBEL_4:def.22">Def22</a></i>;<br/>
<b>take </b><font color="Maroon" title="c8">u</font> = <font color="Maroon" title="c6">y'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c7">z'</font>;<br/>

<a NAME="E8:128"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E5:128"><i><font color="Green" title="E78">A3</font></i></a>, <a class="txt" href="waybel_4.html#E7:128"><i><font color="Green" title="E79">A4</font></i></a>, <a class="ref" href="waybel_4.html#D6" target="_self" title="WAYBEL_4:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E9:128"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font>
 ;<br/>

<a NAME="E10:128"/>
( <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c6">y'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">u</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/>
<b>hence </b><a NAME="E11:128"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c8">u</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E5:128"><i><font color="Green" title="E78">A3</font></i></a>, <a class="ref" href="waybel_4.html#D5" target="_self" title="WAYBEL_4:def.5">Def5</a></i>;<br/>

<a NAME="E12:128"/>
( <font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c7">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">u</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/>
<b>hence </b><a NAME="E13:128"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c8">u</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AR</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E7:128"><i><font color="Green" title="E79">A4</font></i></a>, <a class="ref" href="waybel_4.html#D5" target="_self" title="WAYBEL_4:def.5">Def5</a></i>;<br/>


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