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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</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"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span>;<br/>



<b>set </b><font color="Maroon">B</font> =  <a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">L</font> =  <a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">X</font>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">X</font></span>)</span> ;<br/>
<b>thus </b><a NAME="E2:12"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> implies <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:12_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" target="_self">BoolePoset</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E3:12_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E4:12_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<b>hence </b><a NAME="E5:12_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T2">LATTICE3:2</a></i>;<br/>


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

<b>assume </b><a NAME="E3:12"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E4:12"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T2">LATTICE3:2</a></i>;<br/>
<a NAME="E5:12"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<b>hence </b><a NAME="E6:12"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>


</div>
