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

<b>let </b><font color="Maroon">C</font> be   <a href="waybel_0.html#NM5">Chain</a>;<br/>

<a NAME="E1:26_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> holds  <font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:26_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1_1"/>
( ( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> ) or ( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> ) or ( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> ) or ( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> ) or ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> ) or ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> ) or ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> ) or ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> ) )
 </span><b>by </b><i><a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E48</font></i>: <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E4:26_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E55</font></i>: <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E4:26_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_1_3"/><b>then </b>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/><a NAME="E3:26_1_1_1_1_3"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/><i><font color="Green">E69</font></i>: <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E6:26_1_1_1_1_3"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_4"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E71</font></i>: <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E4:26_1_1_1_1_4"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_5"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E73</font></i>: <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E4:26_1_1_1_1_5"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_6"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_6"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> )
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_1_6"/><b>then </b>
<font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/><a NAME="E3:26_1_1_1_1_6"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/><i><font color="Green">E77</font></i>: <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E6:26_1_1_1_1_6"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_7"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_7"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E79</font></i>: <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E4:26_1_1_1_1_7"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_8"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_8"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E81</font></i>: <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E4:26_1_1_1_1_8"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1_1"/>
<font color="Maroon">C</font> is <a href="waybel_1.html#V2">distributive</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D3">WAYBEL_1:def 3</a></i>;<br/>


</div>
