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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V4">complete</a> <a href="lattice6.html#V2" target="_self">co-noetherian</a> <a href="lattices.html#NM1">Lattice</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span>] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K4">%</a> ;<br/>
<a NAME="E1:55"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 
<b>by </b><i><a class="ref" href="lattice6.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E2:55"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span> ;<br/>
<a NAME="E2:55_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  holds <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></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"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:55_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:55_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>
<a NAME="E4:55_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:55_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <span class="p3">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a>  = 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D6">LATTICE3:def 6</a></i>
<br/>.= 
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D7">LATTICE3:def 7</a></i>
;<br/>
<a NAME="E6:55_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E4:55_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>
<i><font color="Green">E46</font></i>:  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D7">LATTICE3:def 7</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a>  = 
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D6">LATTICE3:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E9:55_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></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"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="lattice6.html#E3:55_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="lattice6.html#E6:55_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="lattice6.html#E7:55_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:55_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:55_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span> 
 ;<br/>

<a NAME="E2:55_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></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"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:55_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:55_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> ]
 <b>by </b><i><a class="txt" href="lattice6.html#E1:55_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_3"/>
(  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="lattice6.html#V3" target="_self">completely-meet-irreducible</a> or  not  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="lattice6.html#V3" target="_self">completely-meet-irreducible</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_3_1"><b>suppose </b></a><a NAME="E1:55_1_3_1"/>
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="lattice6.html#V3" target="_self">completely-meet-irreducible</a>
 ;<br/><div class="add"><a NAME="E2:55_1_3_1"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="lattice6.html#V3" target="_self">completely-meet-irreducible</a> </span>}</span> 
 ;<br/><a NAME="E3:55_1_3_1"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E4:55_1_3_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <span class="p3">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:55_1_3_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <span class="p3">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="lattice3.html#K5">%</a> <span class="p4">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K16">"/\"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <span class="p3">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T43">LATTICE3:43</a></i>
<br/>.= 
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D7">LATTICE3:def 7</a></i>
;<br/><b>hence </b><a NAME="E7:55_1_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="lattice6.html#E5:55_1_3_1"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_3_2"><b>suppose </b></a><a NAME="E1:55_1_3_2"/>
not  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="lattice6.html#V3" target="_self">completely-meet-irreducible</a>
 ;<br/><div class="add"><a NAME="E2:55_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="lattice6.html#K3" target="_self">*'</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice6.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E3:55_1_3_2"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> ;<br/><a NAME="E4:55_1_3_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>4</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:55_1_3_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_1_3_2_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:55_1_3_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="lattice6.html#E3:55_1_3_2_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:55_1_3_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:55_1_3_2_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="lattice6.html#E5:55_1_3_2_1"><i><font color="Green">E46</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:55_1_3_2"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E6:55_1_3_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>4</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  holds <br/> <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_3_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:55_1_3_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_1_3_2_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:55_1_3_2_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="lattice6.html#E3:55_1_3_2_2"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:55_1_3_2_2"/><i><font color="Green">E48</font></i>: 
(  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="lattice6.html#E5:55_1_3_2_2"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E8:55_1_3_2_2"/>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E7:55_1_3_2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<a NAME="E9:55_1_3_2_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:55_1_3_2_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<a NAME="E10:55_1_3_2_2"/>
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#R3">is_less_than</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:55_1_3_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<a NAME="E11:55_1_3_2_2"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E7:55_1_3_2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E12:55_1_3_2_2"/>
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="lattice6.html#E9:55_1_3_2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:55_1_3_2"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#R3">is_less_than</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/><a NAME="E8:55_1_3_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3">is_less_than</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>4</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_3_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:55_1_3_2_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#R3">is_less_than</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:55_1_3_2_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_3_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:55_1_3_2_3_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span> 
 ;<br/>

<a NAME="E2:55_1_3_2_3_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:55_1_3_2_3_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K4">%</a> 
 <b>by </b><i><a class="txt" href="lattice6.html#E3:55_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="lattice6.html#E1:55_1_3_2_3_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<i><font color="Green">E51</font></i>: <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> 
<b>by </b><i><a class="txt" href="lattice6.html#E2:55_1_3_2_3_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="lattice3.html#D6">LATTICE3:def 6</a></i>
<br/>.= 
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="lattice6.html#E4:55_1_3_2_3_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/>
<a NAME="E6:55_1_3_2_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>4</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_3_2_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:55_1_3_2_3_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:55_1_3_2_3_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:55_1_3_2_3_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="lattice6.html#E2:55_1_3_2_3_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a>  = 
<font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D6">LATTICE3:def 6</a></i>
<br/>.= 
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<a NAME="E4:55_1_3_2_3_1_2"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E4:55_1_3_2_3_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>;<br/>
<a NAME="E5:55_1_3_2_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:55_1_3_2_3_1_2"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:55_1_3_2_3_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="lattice6.html#E1:55_1_3_2_3_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:55_1_3_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:55_1_3_2_3_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:55_1_3_2_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<a NAME="E9:55_1_3_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E10:55_1_3_2_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lattice6.html#E5:55_1_3_2_3_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:55_1_3_2_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#R3">is_less_than</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E4:55_1_3_2_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:55_1_3_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:55_1_3_2"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lattice6.html#E7:55_1_3_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/><i><font color="Green">E48</font></i>:  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D7">LATTICE3:def 7</a></i>
;<br/><a NAME="E11:55_1_3_2"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>(  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) </span>}</span>  &amp;  <a href="lattice3.html#K5">%</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>;<br/><b>hence </b><a NAME="E12:55_1_3_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="lattice6.html#E5:55_1_3_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="lattice6.html#E9:55_1_3_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="lattice6.html#E10:55_1_3_2"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:55"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="wellfnd1.html#S3">WELLFND1:sch 3</a>(<a class="txt" href="lattice6.html#E2:55"><i><font color="Green">E40</font></i></a>, <a class="txt" href="lattice6.html#E1:55"><i><font color="Green">E39</font></i></a>);<br/></i>
<a NAME="E4:55"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:55_2"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a>  <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K6">~</a> </span>)</span> #)
 
<b>by </b><i><a class="ref" href="lattice3.html#D5">LATTICE3:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:55_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K4">%</a> 
 <b>by </b><i><a class="txt" href="lattice6.html#E3:55"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E5:55_2"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>;<br/>
<a NAME="E6:55_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:55_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lattice6.html#E4:55_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="lattice6.html#E5:55_2"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:55"/>
 <a href="lattice6.html#K5" target="_self">MIRRS</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattice6.html#V9" target="_self">infimum-dense</a>
 <b>by </b><i><a class="ref" href="lattice6.html#D14" target="_self">Def14</a></i>;<br/>


</div>
