<?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="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"> for <font color="Olive" title="b1">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">D</font> is <a href="lattice6.html#V9" title="LATTICE6:attr.9">infimum-dense</a> iff <font color="Olive" title="b1">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a> )</span><br/><b>let </b><font color="Maroon" title="c2">D</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c2">D</font> is <a href="lattice6.html#V9" title="LATTICE6:attr.9">infimum-dense</a> iff <font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a> )</span><br/>



<div><i><font color="Green" title="E38">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c2">D</font> is <a href="lattice6.html#V9" title="LATTICE6:attr.9">infimum-dense</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a></span><br/><a NAME="E2:51_1"/>
for <font color="Olive" title="b1">a</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>  ex <font color="Olive" title="b2">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> </span>)</span> st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b2">Y</font>,<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="51_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</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">  ex <font color="Olive" title="b1">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> </span>)</span> st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b1">Y</font>,<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>consider </b><font color="Maroon" title="c4">D'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E2:51_1_1"/><i><font color="Green" title="E40">A3</font></i>: 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c4">D'</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E1:51_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="lattice6.html#D14" target="_self" title="LATTICE6:def.14">Def14</a></i>;<br/>
<a NAME="E3:51_1_1"/><i><font color="Green" title="E41">A4</font></i>: 
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c4">D'</font> &amp; ( for <font color="Olive" title="b1">b</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">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c4">D'</font> holds <br/><font color="Olive" title="b1">b</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="c3">a</font> ) )
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:51_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<a NAME="E4:51_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D'</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D'</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:51_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:51_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 
;<br/>
<b>hence </b><a NAME="E3:51_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">D'</font> = <font color="Maroon" title="c4">D'</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:51_1_1"/>
for <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 <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  holds <br/><font color="Maroon" title="c3">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">u</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"> ( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  implies <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">u</font> )</span><br/>

<b>assume </b><a NAME="E1:51_1_1_2"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">u</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">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="E3:51_1_1_2"/><i><font color="Green" title="E42">A5</font></i>: 
( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u'</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  &amp; <font color="Maroon" title="c7">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D'</font> )
 ;<br/>
<a NAME="E4:51_1_1_2"/><i><font color="Green" title="E43">A6</font></i>: 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">u'</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:51_1_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="lattice6.html#E3:51_1_1_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  = 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E6:51_1_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">u</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:51_1_1_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="lattice6.html#E4:51_1_1_2"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</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="E7:51_1_1"/><b>then </b><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E8:51_1_1"/>
for <font color="Olive" title="b1">b</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="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="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"> ( <font color="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c6">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">a</font> )</span><br/>

<b>assume </b><a NAME="E1:51_1_1_3"/><i><font color="Green" title="E43">A8</font></i>: 
<font color="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">b</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">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">a</font></span><br/>

<i><font color="Green" title="E44">A9</font></i>: <font color="Maroon" title="c6">b</font> = 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></i>
;<br/>
<i><font color="Green" title="E45">A10</font></i>: <font color="Maroon" title="c3">a</font> = 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></i>
;<br/>
<a NAME="E4:51_1_1_3"/>
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> <font color="Maroon" title="c5">D'</font> holds <br/> <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c6">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D'</font> implies  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c6">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">u</font> )</span><br/>

<b>assume </b><a NAME="E1:51_1_1_3_3"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D'</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="c6">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">u</font></span><br/>

<a NAME="E2:51_1_1_3_3"/><b>then </b>
<font color="Maroon" title="c7">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">d</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D'</font> </span>}</span> 
 
;<br/>
<a NAME="E3:51_1_1_3_3"/><b>then </b>
<font color="Maroon" title="c6">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">u</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:51_1_1_3"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:51_1_1_3_3"/>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c6">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">u</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E2:51_1_1_3"><i><font color="Green" title="E44">A9</font></i></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</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="E5:51_1_1_3"/><b>then </b>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c6">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c5">D'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<a NAME="E6:51_1_1_3"/><b>then </b>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c6">b</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="c3">a</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:51_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<b>hence </b><a NAME="E7:51_1_1_3"/>
<font color="Maroon" title="c6">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E2:51_1_1_3"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="lattice6.html#E3:51_1_1_3"><i><font color="Green" title="E45">A10</font></i></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</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="E9:51_1_1"/><b>then </b><i><font color="Green" title="E43">A11</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> </span>)</span>,<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#E7:51_1_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/>
<a NAME="E10:51_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  )</span><br/>

<b>assume </b><a NAME="E1:51_1_1_4"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> </span><br/>

<a NAME="E2:51_1_1_4"/><b>then </b>
 ex <font color="Olive" title="b1">x'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  &amp; <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D'</font> )
 
;<br/>
<b>hence </b><a NAME="E3:51_1_1_4"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> 
 ; <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="E11:51_1_1"/><b>then </b>
<font color="Maroon" title="c5">D'</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:51_1_1"/>
 ex <font color="Olive" title="b1">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> </span>)</span> st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Olive" title="b1">Y</font>,<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#E9:51_1_1"><i><font color="Green" title="E43">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><b>hence </b><a NAME="E3:51_1"/>
<font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#T15" title="WAYBEL_6:th.15">WAYBEL_6:15</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="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_2"/><i><font color="Green" title="E39">A12</font></i>: 
<font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">D</font> is <a href="lattice6.html#V9" title="LATTICE6:attr.9">infimum-dense</a></span><br/><a NAME="E2:51_2"/>
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 <font color="Maroon" title="c2">D</font> st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</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="51_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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 <font color="Maroon" title="c2">D</font> st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Olive" title="b1">D'</font>,<font color="Maroon" title="c1">L</font></span><br/>

<b>consider </b><font color="Maroon" title="c4">Y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> </span>)</span><b> such that </b><br/><a NAME="E2:51_2_1"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">Y</font>,<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#E1:51_2"><i><font color="Green" title="E39">A12</font></i></a>, <a class="ref" href="waybel_6.html#T15" title="WAYBEL_6:th.15">WAYBEL_6:15</a></i>;<br/>
<a NAME="E3:51_2_1"/><i><font color="Green" title="E41">A14</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c4">Y</font> &amp; ( for <font color="Olive" title="b1">b</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">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c4">Y</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font> ) )
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:51_2_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/>
<a NAME="E4:51_2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="51_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="E1:51_2_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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/>

<a NAME="E2:51_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> 
 
;<br/>
<b>hence </b><a NAME="E3:51_2_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c4">Y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:51_2_1"/>
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="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1_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="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</font> implies <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q</font> )</span><br/>

<b>assume </b><a NAME="E1:51_2_1_2"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">q'</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="E3:51_2_1_2"/><i><font color="Green" title="E42">A15</font></i>: 
( <font color="Maroon" title="c6">q</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="c7">q'</font> &amp; <font color="Maroon" title="c7">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )
 ;<br/>
<a NAME="E4:51_2_1_2"/><i><font color="Green" title="E43">A16</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">q'</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:51_2_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="lattice6.html#E3:51_2_1_2"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<font color="Maroon" title="c7">q'</font> = 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c7">q'</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c7">q'</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
<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="E6:51_2_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:51_2_1_2"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="lattice6.html#E4:51_2_1_2"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</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="E7:51_2_1"/><b>then </b><i><font color="Green" title="E42">A17</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a>  <a href="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<a NAME="E8:51_2_1"/>
for <font color="Olive" title="b1">b</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">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a>  <a href="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</font> holds <br/><font color="Olive" title="b1">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</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">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a>  <a href="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</font> implies <font color="Maroon" title="c6">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> )</span><br/>

<b>assume </b><a NAME="E1:51_2_1_3"/><i><font color="Green" title="E43">A18</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a>  <a href="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</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">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font></span><br/>

<a NAME="E2:51_2_1_3"/>
for <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 <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> holds <br/><font color="Maroon" title="c6">b</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="Olive" title="b1">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">u</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"> ( <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> implies <font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">u</font> )</span><br/>

<b>assume </b><a NAME="E1:51_2_1_3_1"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</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">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">u</font></span><br/>

<a NAME="E2:51_2_1_3_1"/><b>then </b>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Olive" title="b1">d</font></span>)</span> where <font color="Olive" title="b1">d</font> is   <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> : <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> </span>}</span> 
 
;<br/>
<a NAME="E3:51_2_1_3_1"/><b>then </b><i><font color="Green" title="E44">A19</font></i>: 
<font color="Maroon" title="c6">b</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="c7">u</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:51_2_1_3"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  = 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c7">u</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c7">u</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E5:51_2_1_3_1"/>
<font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">u</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:51_2_1_3_1"><i><font color="Green" title="E44">A19</font></i></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</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="E3:51_2_1_3"/><b>then </b>
<font color="Maroon" title="c6">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E4:51_2_1_3"/><b>then </b>
<font color="Maroon" title="c6">b</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="c3">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:51_2_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/>
<b>hence </b><a NAME="E5:51_2_1_3"/>
<font color="Maroon" title="c6">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</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="E9:51_2_1"/><b>then </b><i><font color="Green" title="E43">A20</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">(<span class="default"><a href="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E7:51_2_1"><i><font color="Green" title="E42">A17</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<a NAME="E10:51_2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> )</span><br/>

<b>assume </b><a NAME="E1:51_2_1_4"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">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><b> such that </b><br/><a NAME="E3:51_2_1_4"/><i><font color="Green" title="E44">A21</font></i>: 
( <font color="Maroon" title="c6">x</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="c7">x'</font> &amp; <font color="Maroon" title="c7">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )
 ;<br/>
<a NAME="E4:51_2_1_4"/>
<font color="Maroon" title="c7">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:51_2_1_4"><i><font color="Green" title="E44">A21</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">y</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="E6:51_2_1_4"/><i><font color="Green" title="E45">A22</font></i>: 
( <font color="Maroon" title="c7">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> )
 ;<br/>
<font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c8">y</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
<b>by </b><i><a class="txt" href="lattice6.html#E3:51_2_1_4"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="lattice6.html#E6:51_2_1_4"><i><font color="Green" title="E45">A22</font></i></a>, <a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c8">y</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="E8:51_2_1_4"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E6:51_2_1_4"><i><font color="Green" title="E45">A22</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="E11:51_2_1"/><b>then </b>
 <a href="lattice6.html#K2" title="LATTICE6:func.2">%</a> <font color="Maroon" title="c5">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:51_2_1"/>
 ex <font color="Olive" title="b1">D'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">D</font> st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</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#E9:51_2_1"><i><font color="Green" title="E43">A20</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="E3:51_2"/>
<font color="Maroon" title="c2">D</font> is <a href="lattice6.html#V9" title="LATTICE6:attr.9">infimum-dense</a>
 <b>by </b><i><a class="ref" href="lattice6.html#D14" target="_self" title="LATTICE6:def.14">Def14</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="E3:51"/>
( <font color="Maroon" title="c2">D</font> is <a href="lattice6.html#V9" title="LATTICE6:attr.9">infimum-dense</a> iff <font color="Maroon" title="c2">D</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a>  is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a> )
 <b>by </b><i><a class="txt" href="lattice6.html#E1:51"><i><font color="Green" title="E38">A1</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>
