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

<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>



<b>set </b><font color="Maroon" title="c4">EL</font> =  <a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>set </b><font color="Maroon" title="c5">El</font> =  <a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c2">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">b'</font> = <font color="Maroon" title="c3">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>thus </b><a NAME="E3:9"/>
( <font color="Maroon" title="c2">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon" title="c2">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">b</font>
 ;<br/>

<a NAME="E2:9_1"/><b>then </b>
<font color="Maroon" title="c6">a'</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">b'</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
;<br/>
<a NAME="E3:9_1"/><b>then </b>
<font color="Maroon" title="c6">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">b'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1"/>
<font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="lattice5.html#T5" target="_self" title="LATTICE5:th.5">Th5</a></i>;<br/>


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

<b>thus </b><a NAME="E4:9"/>
( <font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c2">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_2"/>
<font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">b</font>
 ;<br/>

<a NAME="E2:9_2"/><b>then </b>
<font color="Maroon" title="c6">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">b'</font>
 
<b>by </b><i><a class="ref" href="lattice5.html#T5" target="_self" title="LATTICE5:th.5">Th5</a></i>;<br/>
<a NAME="E3:9_2"/><b>then </b>
<font color="Maroon" title="c6">a'</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">b'</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></i>;<br/>
<b>hence </b><a NAME="E4:9_2"/>
<font color="Maroon" title="c2">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">b</font>
 ;<br/>


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


</div>
