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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>thus </b><a NAME="E1:18"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="yellow11.html#V1" target="_self">modular</a> implies for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:18_1_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E2:18_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E3:18_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E2:18_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="yellow11.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E4:18_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E2:18_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><b>hence </b><a NAME="E5:18_1_1"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="yellow11.html#V1" target="_self">modular</a>
 <b>by </b><i><a class="txt" href="yellow11.html#E1:18_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E2:18_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow11.html#E3:18_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow11.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="yellow11.html#V1" target="_self">modular</a> implies for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) )
 ;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="yellow11.html#V1" target="_self">modular</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:18_2"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="yellow11.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E4:18_2"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow11.html#T8" target="_self">Th8</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E5:18_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E6:18_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_3.html#T2">YELLOW_3:2</a></i>;<br/><a NAME="E7:18_2"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>;<br/><a NAME="E8:18_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_5.html#T7">YELLOW_5:7</a></i>;<br/><a NAME="E9:18_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T17">LATTICE3:17</a></i>;<br/><a NAME="E10:18_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_3.html#T2">YELLOW_3:2</a></i>;<br/><a NAME="E11:18_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="yellow11.html#E1:18_2_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow11.html#E1:18_2_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
;<br/><b>hence </b><a NAME="E3:18_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E12:18_2"/><i><font color="Green">E19</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E13:18_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a></i>;<br/><a NAME="E14:18_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_3.html#T2">YELLOW_3:2</a></i>;<br/><a NAME="E15:18_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#T18">LATTICE3:18</a></i>;<br/><a NAME="E16:18_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T7">YELLOW_5:7</a></i>;<br/><b>thus </b><a NAME="E17:18_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<b>thus </b><a NAME="E2:18_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_2_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:18_2_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E3:18_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:18_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:18_2_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_5.html#T8">YELLOW_5:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E3:18_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_2_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:18_2_2_2_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E3:18_2_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T9">YELLOW_5:9</a></i>;<br/><b>hence </b><a NAME="E4:18_2_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:18_2_2_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E4:18_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_2_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E2:18_2_2_3_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E3:18_2_2_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow11.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E4:18_2_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E2:18_2_2_3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E5:18_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E11:18_2"><i><font color="Green">E18</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:18_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_2_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:18_2_2_4_1"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a>, <a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>;<br/><font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>12</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T17">LATTICE3:17</a></i>
;<br/><b>hence </b><a NAME="E4:18_2_2_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E7:18_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow11.html#E10:18_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="yellow11.html#E11:18_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow11.html#E2:18_2_2_4_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_2_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E7:18_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:18_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_2_5_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:18_2_2_5_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E3:18_2_2_5_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow11.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E4:18_2_2_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E2:18_2_2_5_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_2_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E9:18_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_2_6_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E2:18_2_2_6_1"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_5.html#T1">YELLOW_5:1</a>, <a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>;<br/><font color="Maroon">c<sub>12</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T18">LATTICE3:18</a></i>
;<br/><b>hence </b><a NAME="E4:18_2_2_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E11:18_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow11.html#E12:18_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="yellow11.html#E16:18_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow11.html#E2:18_2_2_6_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_2_6"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E10:18_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_2_7_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E2:18_2_2_7_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E3:18_2_2_7_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_3.html#T2">YELLOW_3:2</a></i>;<br/><a NAME="E4:18_2_2_7_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>;<br/><b>hence </b><a NAME="E5:18_2_2_7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:18_2_2_7_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_5.html#T8">YELLOW_5:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_2_7"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E11:18_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_2_8_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E2:18_2_2_8_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E3:18_2_2_8_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:18_2_2_8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow11.html#E1:18_2_2_8_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_2_2_8"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>


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

<a NAME="E12:18_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<b>hence </b><a NAME="E13:18_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>;<br/>

<a NAME="E14:18_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E15:18_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>;<br/>

<a NAME="E16:18_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<b>hence </b><a NAME="E17:18_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>;<br/>

<a NAME="E18:18_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E19:18_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>;<br/>

<b>thus </b><a NAME="E20:18_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E7:18_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow11.html#E10:18_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>

<b>thus </b><a NAME="E21:18_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E4:18_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>;<br/>

<b>thus </b><font color="Maroon">c<sub>12</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T18">LATTICE3:18</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>12</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T17">LATTICE3:17</a></i>
;<br/>

<a NAME="E24:18_2_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E25:18_2_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T9">YELLOW_5:9</a></i>;<br/>
<a NAME="E26:18_2_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow11.html#E3:18_2"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E27:18_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T2">YELLOW_3:2</a></i>;<br/>
<a NAME="E28:18_2_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T18">LATTICE3:18</a></i>;<br/>
<a NAME="E29:18_2_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T7">YELLOW_5:7</a></i>;<br/>
<b>hence </b><a NAME="E30:18_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E25:18_2_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) ) implies <font color="Maroon">c<sub>1</sub></font> is <a href="yellow11.html#V1" target="_self">modular</a> )
 ;<br/>


</div>
