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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <a href="lattice6.html#V1" title="LATTICE6:attr.1">noetherian</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font> is <a href="lattice6.html#V8" title="LATTICE6:attr.8">supremum-dense</a></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>] <b>means </b> ex <font color="Olive" title="b1">D'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> st  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">D'</font>,<font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E1:57"/><i><font color="Green" title="E40">A1</font></i>: 
 <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font> is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 
<b>by </b><i><a class="ref" href="lattice6.html#D3" target="_self" title="LATTICE6:def.3">Def3</a></i>;<br/>
<a NAME="E2:57"/><i><font color="Green" title="E41">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st ( for <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>] )</span><br/>

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green" title="E42">A3</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>]</span><br/>

<b>set </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span> ;<br/>
<a NAME="E2:57_1"/><i><font color="Green" title="E43">A4</font></i>: 
for <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">d</font> where <font color="Olive" title="b2">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  holds <br/>( <font color="Olive" title="b1">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  implies ( <font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E1:57_1_1"/>
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> )</span><br/>

<b>then consider </b><font color="Maroon" title="c5">u'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:57_1_1"/><i><font color="Green" title="E44">A5</font></i>: 
( <font color="Maroon" title="c5">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> &amp; <font color="Maroon" title="c5">u'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">u'</font> )
 ;<br/>
<a NAME="E4:57_1_1"/><i><font color="Green" title="E45">A6</font></i>: 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>;<br/>
<a NAME="E5:57_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></i>;<br/>
<a NAME="E6:57_1_1"/><b>then </b>
<font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:57_1_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></i>;<br/>
<b>hence </b><a NAME="E7:57_1_1"/>
( <font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 <b>by </b><i><a class="txt" href="lattice6.html#E3:57_1_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="lattice6.html#E4:57_1_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:57_1"/><i><font color="Green" title="E44">A7</font></i>: 
for <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">d</font> where <font color="Olive" title="b2">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> ] )</span><br/>

<b>assume </b><a NAME="E1:57_1_2"/>
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> ]</span><br/>

<a NAME="E2:57_1_2"/><b>then </b>
( <font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:57_1"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:57_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> ]
 <b>by </b><i><a class="txt" href="lattice6.html#E1:57_1"><i><font color="Green" title="E42">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_3"/>
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a> or  not  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_3_1"><b>suppose </b></a><a NAME="E1:57_1_3_1"/>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>]</span><br/><div class="add"><a NAME="E2:57_1_3_1"/><b>then </b>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">a</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a> </span>}</span> 
 ;<br/><a NAME="E3:57_1_3_1"/><b>then </b>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:57_1_3_1"/><b>then </b><i><font color="Green" title="E45">A8</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E5:57_1_3_1"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T43" title="LATTICE3:th.43">LATTICE3:43</a></i>;<br/><b>hence </b><a NAME="E6:57_1_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>]
 <b>by </b><i><a class="txt" href="lattice6.html#E4:57_1_3_1"><i><font color="Green" title="E45">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_3_2"><b>suppose </b></a><a NAME="E1:57_1_3_2"/>
not  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>]</span><br/><div class="add"><a NAME="E2:57_1_3_2"/><b>then </b><i><font color="Green" title="E45">A9</font></i>: 
 <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="lattice6.html#D9" target="_self" title="LATTICE6:def.9">Def9</a></i>;<br/><b>set </b><font color="Maroon" title="c4">G</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c5">D'</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span> ;<br/><a NAME="E3:57_1_3_2"/>
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">H</font> where <font color="Olive" title="b2">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">d</font> where <font color="Olive" title="b4">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b4">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b2">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  holds <br/><font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  implies <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:57_1_3_2_1"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">v'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:57_1_3_2_1"/><i><font color="Green" title="E46">A10</font></i>: 
( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v'</font> &amp; <font color="Maroon" title="c7">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">H</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span><b> such that </b><br/><a NAME="E5:57_1_3_2_1"/><i><font color="Green" title="E47">A11</font></i>: 
( <font color="Maroon" title="c8">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v'</font> &amp;  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">d</font> where <font color="Olive" title="b2">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c8">H</font>,<font color="Maroon" title="c1">L</font> ) )
 <b>by </b><i><a class="txt" href="lattice6.html#E3:57_1_3_2_1"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:57_1_3_2_1"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:57_1_3_2_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="lattice6.html#E5:57_1_3_2_1"><i><font color="Green" title="E47">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E4:57_1_3_2"/><b>then </b><i><font color="Green" title="E46">A12</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E5:57_1_3_2"/>
for <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">H</font> where <font color="Olive" title="b2">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">d</font> where <font color="Olive" title="b4">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b4">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b2">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  holds <br/><font color="Olive" title="b1">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_3_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  implies <font color="Maroon" title="c6">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> )</span><br/>

<b>assume </b><a NAME="E1:57_1_3_2_2"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:57_1_3_2_2"/><i><font color="Green" title="E47">A13</font></i>: 
( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">h</font> &amp; <font color="Maroon" title="c7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">h'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span><b> such that </b><br/><a NAME="E5:57_1_3_2_2"/><i><font color="Green" title="E48">A14</font></i>: 
( <font color="Maroon" title="c8">h'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> &amp;  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">d</font> where <font color="Olive" title="b2">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c8">h'</font>,<font color="Maroon" title="c1">L</font> ) )
 <b>by </b><i><a class="txt" href="lattice6.html#E3:57_1_3_2_2"><i><font color="Green" title="E47">A13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span><b> such that </b><br/><a NAME="E7:57_1_3_2_2"/><i><font color="Green" title="E49">A15</font></i>: 
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c7">h</font>,<font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="lattice6.html#E5:57_1_3_2_2"><i><font color="Green" title="E48">A14</font></i></a></i>;<br/>
<a NAME="E8:57_1_3_2_2"/>
<font color="Maroon" title="c7">h</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c9">u</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E7:57_1_3_2_2"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E9:57_1_3_2_2"/><b>then </b><i><font color="Green" title="E50">A16</font></i>: 
<font color="Maroon" title="c6">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c9">u</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:57_1_3_2_2"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<a NAME="E10:57_1_3_2_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:57_1_3_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E11:57_1_3_2_2"/><b>then </b>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c9">u</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E7:57_1_3_2_2"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E12:57_1_3_2_2"/>
<font color="Maroon" title="c6">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E9:57_1_3_2_2"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E6:57_1_3_2"/><b>then </b><i><font color="Green" title="E47">A17</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/><a NAME="E7:57_1_3_2"/>
for <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">H</font> where <font color="Olive" title="b2">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">d</font> where <font color="Olive" title="b4">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b4">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b2">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">r</font> holds <br/> <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_3_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c6">r</font> implies  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">r</font> )</span><br/>

<b>assume </b><a NAME="E1:57_1_3_2_3"/><i><font color="Green" title="E48">A18</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c6">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">r</font></span><br/>

<a NAME="E2:57_1_3_2_3"/>
for <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">d</font> where <font color="Olive" title="b2">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  holds <br/><font color="Olive" title="b1">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_3_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  implies <font color="Maroon" title="c7">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">r</font> )</span><br/>

<b>assume </b><a NAME="E1:57_1_3_2_3_1"/><i><font color="Green" title="E49">A19</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">r</font></span><br/>

<a NAME="E2:57_1_3_2_3_1"/><i><font color="Green" title="E50">A20</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">D</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span><b> such that </b><br/><a NAME="E4:57_1_3_2_3_1"/><i><font color="Green" title="E51">A21</font></i>: 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:57_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="lattice6.html#E1:57_1_3_2_3_1"><i><font color="Green" title="E49">A19</font></i></a></i>;<br/>
<a NAME="E5:57_1_3_2_3_1"/><i><font color="Green" title="E52">A22</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c8">D</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:57_1_3_2_3_1"><i><font color="Green" title="E50">A20</font></i></a>, <a class="txt" href="lattice6.html#E4:57_1_3_2_3_1"><i><font color="Green" title="E51">A21</font></i></a>, <a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>;<br/>
<a NAME="E6:57_1_3_2_3_1"/>
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D</font> holds <br/><font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">H</font> where <font color="Olive" title="b2">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">d</font> where <font color="Olive" title="b4">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b4">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b2">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_3_2_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D</font> implies <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:57_1_3_2_3_1_1"/><i><font color="Green" title="E53">A23</font></i>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span> </span><br/>

<a NAME="E2:57_1_3_2_3_1_1"/>
<font color="Maroon" title="c8">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:57_1_3_2_3_1"><i><font color="Green" title="E49">A19</font></i></a>, <a class="txt" href="lattice6.html#E4:57_1_3_2_3_1"><i><font color="Green" title="E51">A21</font></i></a>, <a class="txt" href="lattice6.html#E5:57_1_3_2_3_1"><i><font color="Green" title="E52">A22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:57_1_3_2_3_1_1"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="lattice6.html#E1:57_1_3_2_3_1_1"><i><font color="Green" title="E53">A23</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:57_1_3_2_3_1"/><b>then </b>
<font color="Maroon" title="c8">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:57_1_3_2_3_1"/><b>then </b>
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D</font> holds <br/><font color="Olive" title="b1">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">r</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:57_1_3_2_3"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<a NAME="E9:57_1_3_2_3_1"/><b>then </b>
<font color="Maroon" title="c8">D</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c6">r</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E10:57_1_3_2_3_1"/>
<font color="Maroon" title="c7">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E5:57_1_3_2_3_1"><i><font color="Green" title="E52">A22</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:57_1_3_2_3"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c6">r</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E4:57_1_3_2_3"/>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E2:57_1_3_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E8:57_1_3_2"/><b>then </b>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> :  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">d</font> where <font color="Olive" title="b3">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b3">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> ) </span>}</span>  &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">H</font>,<font color="Maroon" title="c1">L</font> ) </span>}</span> </span>)</span>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E6:57_1_3_2"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/><b>hence </b><a NAME="E9:57_1_3_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>]
 <b>by </b><i><a class="txt" href="lattice6.html#E4:57_1_3_2"><i><font color="Green" title="E46">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:57"/><i><font color="Green" title="E42">A24</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>]
 
<b>from </b><i><a class="ref" href="wellfnd1.html#S3" title="WELLFND1:sch.3">WELLFND1:sch 3</a>(<a class="txt" href="lattice6.html#E2:57"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="lattice6.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>);<br/></i>
<a NAME="E4:57"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b2">D'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b2">D'</font>,<font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">D'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">D'</font>,<font color="Maroon" title="c1">L</font></span><br/>

 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E2:57_2"/>
 ex <font color="Olive" title="b1">D'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font></span>)</span> st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Olive" title="b1">D'</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:57"><i><font color="Green" title="E42">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:57"/>
 <a href="lattice6.html#K6" title="LATTICE6:func.6">JIRRS</a> <font color="Maroon" title="c1">L</font> is <a href="lattice6.html#V8" title="LATTICE6:attr.8">supremum-dense</a>
 <b>by </b><i><a class="ref" href="lattice6.html#D13" target="_self" title="LATTICE6:def.13">Def13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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