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

<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>

<a NAME="E1:48"/><i><font color="Green">E18</font></i>: 
(  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="filter_1.html#K9">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="filter_1.html#K9">]</a></span> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">p</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q</font> </span>}</span>  &amp;  <a href="lattice3.html#K2" target="_self">LattRel</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="filter_1.html#K9">[</a><span class="default"><font color="Olive">p'</font>,<font color="Olive">q'</font></span><a href="filter_1.html#K9">]</a></span> where <font color="Olive">p'</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>, <font color="Olive">q'</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> : <font color="Olive">p'</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q'</font> </span>}</span>  &amp; <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">L</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">L</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">L</font>) #) &amp;  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> #) <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">L</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">L</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">L</font>) #) )
 
<b>by </b><i><a class="ref" href="filter_1.html#D8">FILTER_1:def 8</a>, <a class="ref" href="lattice2.html#D2">LATTICE2:def 2</a></i>;<br/>
<b>thus </b><a NAME="E2:48"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K6" target="_self">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<b>thus </b><a NAME="E1:48_1"/>
( <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> <span class="p1">(<span class="default"><a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K6" target="_self">~</a>  implies <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>  <a href="lattice3.html#K2" target="_self">LattRel</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1_1"/>
<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> <span class="p1">(<span class="default"><a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K6" target="_self">~</a> 
 ;<br/>

<a NAME="E2:48_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:48_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="filter_1.html#K9">]</a></span> &amp; <font color="Maroon">p</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> <b>by </b><i/>;<br/>
<a NAME="E6:48_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p'</font> )
 
<b>by </b><i>, <a class="ref" href="lattice2.html#T53">LATTICE2:53</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E7:48_1_1"/>
<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>  <a href="lattice3.html#K2" target="_self">LattRel</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E2:48_1"/>
<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>  <a href="lattice3.html#K2" target="_self">LattRel</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">p'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>, <font color="Maroon">q'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span><b> such that </b><br/><a NAME="E4:48_1"/><i><font color="Green">E25</font></i>: 
( <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#R1">=</a> <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">p'</font>,<font color="Maroon">q'</font></span><a href="filter_1.html#K9">]</a></span> &amp; <font color="Maroon">p'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q'</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p'</font>, <font color="Maroon">q</font> = <font color="Maroon">q'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E6:48_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="lattice2.html#T54">LATTICE2:54</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E7:48_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:48_1"/>
<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> <span class="p1">(<span class="default"><a href="lattice3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K6" target="_self">~</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>


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

<b>hence </b><a NAME="E3:48"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K3" target="_self">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K7" target="_self">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K3" target="_self">LattPOSet</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>
 <b>by </b><i/>;<br/>


</div>
