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

<b>let </b><font color="Maroon">L</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>thus </b><a NAME="E1:6"/>
( <font color="Maroon">L</font> is <a href="orders_2.html#V4">antisymmetric</a> implies <font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is <a href="orders_2.html#V4">antisymmetric</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">L</font> is <a href="orders_2.html#V4">antisymmetric</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a><br/>

<b>assume </b><a NAME="E2:6_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&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">y</font> )
 ;<br/>

<a NAME="E3:6_1"/><b>then </b>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">y</font> &amp;  <a href="lattice3.html#K9">~</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">y</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:6_1"/><b>then </b>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9">~</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/>
<b>hence </b><a NAME="E5:6_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>


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

<b>assume </b><a NAME="E2:6"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is <a href="orders_2.html#V4">antisymmetric</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a><br/>

<b>assume </b><a NAME="E3:6"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&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">y</font> )
 ;<br/>

<a NAME="E4:6"/><b>then </b>
( <font color="Maroon">x</font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="lattice3.html#K8">~</a>  &amp; <font color="Maroon">x</font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="lattice3.html#K8">~</a>  )
 
<b>by </b><i><a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>
<a NAME="E5:6"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="lattice3.html#K8">~</a> 
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:6"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>


</div>
