<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<b>thus </b><a NAME="E1:17"/>
( <font color="Maroon" title="c1">L</font> is <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b4">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font>, <font color="Maroon" title="c6">e</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><a NAME="E2:17_1_1"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">e</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">e</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">e</font> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">e</font> )
 <b>and </b><br/><a NAME="E3:17_1_1"/><i><font color="Green" title="E15">A2</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c5">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c3">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c3">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e</font> )
 ;<br/><a NAME="E4:17_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_1_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="yellow11.html#T5" target="_self" title="YELLOW11:th.5">Th5</a></i>;<br/><a NAME="E5:17_1_1"/>
<font color="Maroon" title="c3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_1_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><b>hence </b><a NAME="E6:17_1_1"/>
not <font color="Maroon" title="c1">L</font> is <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a>
 <b>by </b><i><a class="txt" href="yellow11.html#E2:17_1_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="yellow11.html#E3:17_1_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="yellow11.html#E4:17_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="yellow11.html#D3" target="_self" title="YELLOW11:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_1"/>
( <font color="Maroon" title="c1">L</font> is <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b4">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> ) )
 ;<br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2"/>
not <font color="Maroon" title="c1">L</font> is <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">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="E3:17_2"/><i><font color="Green" title="E14">A4</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> )
 <b>by </b><i><a class="ref" href="yellow11.html#D3" target="_self" title="YELLOW11:def.3">Def3</a></i>;<br/><a NAME="E4:17_2"/><i><font color="Green" title="E15">A5</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="yellow11.html#T8" target="_self" title="YELLOW11:th.8">Th8</a></i>;<br/><b>set </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>;<br/><b>set </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c3">y</font>;<br/><b>set </b><font color="Maroon" title="c8">z1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>;<br/><b>set </b><font color="Maroon" title="c9">t</font> = <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>;<br/><a NAME="E5:17_2"/>
( <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> &amp; <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E6:17_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="yellow_3.html#T2" title="YELLOW_3:th.2">YELLOW_3:2</a></i>;<br/><a NAME="E7:17_2"/><b>then </b><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>;<br/><a NAME="E8:17_2"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">YELLOW_5:7</a></i>;<br/><a NAME="E9:17_2"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T17" title="LATTICE3:th.17">LATTICE3:17</a></i>;<br/><a NAME="E10:17_2"/><b>then </b><i><font color="Green" title="E17">A7</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="yellow_3.html#T2" title="YELLOW_3:th.2">YELLOW_3:2</a></i>;<br/><a NAME="E11:17_2"/><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_1_1"/><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
 ;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> = 
<font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T16" title="LATTICE3:th.16">LATTICE3:16</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="yellow11.html#E1:17_2_1_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow11.html#E1:17_2_1_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E3:17_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_1"/>
<font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E12:17_2"/><i><font color="Green" title="E19">A10</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E13:17_2"/>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/><a NAME="E14:17_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="yellow_3.html#T2" title="YELLOW_3:th.2">YELLOW_3:2</a></i>;<br/><a NAME="E15:17_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T18" title="LATTICE3:th.18">LATTICE3:18</a></i>;<br/><a NAME="E16:17_2"/><b>then </b><i><font color="Green" title="E20">A11</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">YELLOW_5:7</a></i>;<br/><b>thus </b><a NAME="E17:17_2"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> &amp; <font color="Olive" title="b4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> &amp; <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b4">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> &amp; <font color="Olive" title="b3">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>, <font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>, <font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c13">z1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>, <font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c10">b</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c11">x1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c12">y1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c13">z1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c14">t</font>
;<br/>

<b>thus </b><a NAME="E2:17_2_2"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_2_1_1"/><i><font color="Green" title="E21">A12</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1</font>
 ;<br/><a NAME="E2:17_2_2_1_1"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E3:17_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:17_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="yellow11.html#E1:17_2_2_1_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="yellow_5.html#T8" title="YELLOW_5:th.8">YELLOW_5:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_2_1"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x1</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:17_2_2"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_2_2_1"/><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y1</font>
 ;<br/><a NAME="E2:17_2_2_2_1"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E3:17_2_2_2_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T9" title="YELLOW_5:th.9">YELLOW_5:9</a></i>;<br/><b>hence </b><a NAME="E4:17_2_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="yellow11.html#E1:17_2_2_2_1"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="yellow_5.html#T10" title="YELLOW_5:th.10">YELLOW_5:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_2_2"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y1</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:17_2_2"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">z1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_2_3_1"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z1</font>
 ;<br/><a NAME="E2:17_2_2_3_1"/><b>then </b><i><font color="Green" title="E21">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E3:17_2_2_3_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="yellow11.html#T8" target="_self" title="YELLOW11:th.8">Th8</a></i>;<br/><b>hence </b><a NAME="E4:17_2_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="yellow11.html#E2:17_2_2_3_1"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_2_3"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">z1</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:17_2_2"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">t</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E11:17_2"><i><font color="Green" title="E18">A8</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:17_2_2"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_2_4_1"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y1</font>
 ;<br/><a NAME="E2:17_2_2_4_1"/><b>then </b><i><font color="Green" title="E21">A15</font></i>: 
( <font color="Maroon" title="c11">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1</font> &amp; <font color="Maroon" title="c11">x1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1</font> )
 <b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a>, <a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>;<br/><font color="Maroon" title="c11">x1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">y1</font> = 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c14">t</font>
<b>by </b><i><a class="ref" href="lattice3.html#T17" title="LATTICE3:th.17">LATTICE3:17</a></i>
;<br/><b>hence </b><a NAME="E4:17_2_2_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E7:17_2"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="yellow11.html#E10:17_2"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="yellow11.html#E11:17_2"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="yellow11.html#E2:17_2_2_4_1"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_2_4"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y1</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:17_2_2"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">z1</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:17_2_2"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_2_5_1"/>
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1</font>
 ;<br/><a NAME="E2:17_2_2_5_1"/><b>then </b><i><font color="Green" title="E21">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E3:17_2_2_5_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="yellow11.html#T8" target="_self" title="YELLOW11:th.8">Th8</a></i>;<br/><b>hence </b><a NAME="E4:17_2_2_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="yellow11.html#E2:17_2_2_5_1"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_2_5"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">t</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E9:17_2_2"/>
<font color="Maroon" title="c12">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">z1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_2_6_1"/>
<font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z1</font>
 ;<br/><a NAME="E2:17_2_2_6_1"/><b>then </b><i><font color="Green" title="E21">A17</font></i>: 
( <font color="Maroon" title="c13">z1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z1</font> &amp; <font color="Maroon" title="c13">z1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z1</font> )
 <b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a>, <a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>;<br/><font color="Maroon" title="c12">y1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">z1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
<b>by </b><i><a class="ref" href="lattice3.html#T16" title="LATTICE3:th.16">LATTICE3:16</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T18" title="LATTICE3:th.18">LATTICE3:18</a></i>
;<br/><b>hence </b><a NAME="E4:17_2_2_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E11:17_2"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="yellow11.html#E12:17_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="yellow11.html#E16:17_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="yellow11.html#E2:17_2_2_6_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_2_6"/>
<font color="Maroon" title="c12">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">z1</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E10:17_2_2"/>
<font color="Maroon" title="c12">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_2_7_1"/><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">t</font>
 ;<br/><a NAME="E2:17_2_2_7_1"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/><a NAME="E3:17_2_2_7_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="yellow_3.html#T2" title="YELLOW_3:th.2">YELLOW_3:2</a></i>;<br/><a NAME="E4:17_2_2_7_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>;<br/><b>hence </b><a NAME="E5:17_2_2_7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="yellow11.html#E1:17_2_2_7_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="yellow_5.html#T8" title="YELLOW_5:th.8">YELLOW_5:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_2_7"/>
<font color="Maroon" title="c12">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">t</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E11:17_2_2"/>
<font color="Maroon" title="c13">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2_2_8_1"/><i><font color="Green" title="E21">A19</font></i>: 
<font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">t</font>
 ;<br/><a NAME="E2:17_2_2_8_1"/><b>then </b>
( <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E3:17_2_2_8_1"/><b>then </b>
<font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:17_2_2_8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="yellow11.html#E1:17_2_2_8_1"><i><font color="Green" title="E21">A19</font></i></a>, <a class="ref" href="yellow_5.html#T10" title="YELLOW_5:th.10">YELLOW_5:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_2_2_8"/>
<font color="Maroon" title="c13">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">t</font>
 ;<br/>


</div><b>end;</b></div>

<a NAME="E12:17_2_2"/>
<font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">x1</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="E13:17_2_2"/>
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T10" title="YELLOW_5:th.10">YELLOW_5:10</a></i>;<br/>

<a NAME="E14:17_2_2"/>
<font color="Maroon" title="c10">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">y1</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E15:17_2_2"/>
<font color="Maroon" title="c10">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T10" title="YELLOW_5:th.10">YELLOW_5:10</a></i>;<br/>

<a NAME="E16:17_2_2"/>
<font color="Maroon" title="c12">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">t</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="E17:17_2_2"/>
<font color="Maroon" title="c12">y1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y1</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T10" title="YELLOW_5:th.10">YELLOW_5:10</a></i>;<br/>

<a NAME="E18:17_2_2"/>
<font color="Maroon" title="c13">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">t</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E19:17_2_2"/>
<font color="Maroon" title="c13">z1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z1</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T10" title="YELLOW_5:th.10">YELLOW_5:10</a></i>;<br/>

<b>thus </b><a NAME="E20:17_2_2"/>
<font color="Maroon" title="c11">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E7:17_2"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="yellow11.html#E10:17_2"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/>

<b>thus </b><a NAME="E21:17_2_2"/>
<font color="Maroon" title="c11">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x1</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E4:17_2"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="yellow_5.html#T10" title="YELLOW_5:th.10">YELLOW_5:10</a></i>;<br/>

<b>thus </b><font color="Maroon" title="c12">y1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">z1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
<b>by </b><i><a class="ref" href="lattice3.html#T16" title="LATTICE3:th.16">LATTICE3:16</a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T18" title="LATTICE3:th.18">LATTICE3:18</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c11">x1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">y1</font> = 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon" title="c14">t</font>
<b>by </b><i><a class="ref" href="lattice3.html#T17" title="LATTICE3:th.17">LATTICE3:17</a></i>
;<br/>

<a NAME="E24:17_2_2"/>
( <font color="Maroon" title="c13">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c12">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<a NAME="E25:17_2_2"/><b>then </b><i><font color="Green" title="E21">A20</font></i>: 
<font color="Maroon" title="c12">y1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c13">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T9" title="YELLOW_5:th.9">YELLOW_5:9</a></i>;<br/>
<a NAME="E26:17_2_2"/>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="yellow11.html#E3:17_2"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/>
<a NAME="E27:17_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T2" title="YELLOW_3:th.2">YELLOW_3:2</a></i>;<br/>
<a NAME="E28:17_2_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">z</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T18" title="LATTICE3:th.18">LATTICE3:18</a></i>;<br/>
<a NAME="E29:17_2_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">z1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c12">y1</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T7" title="YELLOW_5:th.7">YELLOW_5:7</a></i>;<br/>
<b>hence </b><a NAME="E30:17_2_2"/>
<font color="Maroon" title="c12">y1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c13">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">t</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E25:17_2_2"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17"/>
( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b4">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> ) ) implies <font color="Maroon" title="c1">L</font> is <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> )
 ;<br/>


</div>
