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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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><span class="lab"><font color="Green" title="E37">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:50_1"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:50_1"/>
 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">D9</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">D9</font>,<font color="Maroon" title="c1">L</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">D9</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">D9</font>,<font color="Maroon" title="c1">L</font>)</span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:50_1_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="waybel_6.html#T15" title="WAYBEL_6:th.15">WAYBEL_6:15</a></span>;<br/>
<a NAME="E3:50_1_1"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
 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="50_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:50_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">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:50_1_1_1"/><span class="kw">then </span>
<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/>
<span class="kw">hence </span><a NAME="E3:50_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 <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:50_1_1"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E6:50_1_1"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
 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="50_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:50_1_1_2"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:50_1_1_2"/>
 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="50_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:50_1_1_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:50_1_1_2_1"/><span class="kw">then </span>
 <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="E3:50_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A8</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1_2"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></span>;<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c7">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:50_1_1_2_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1_2_1"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:50_1_1_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></span>;<br/>
<a NAME="E4:50_1_1_2"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:50_1_1_2"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:50_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>  <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="50_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:50_1_1_3"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x9</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><span class="kw"> such that </span><br/><a NAME="E3:50_1_1_3"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<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">x9</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1_3"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<font color="Maroon" title="c7">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 ;<br/>
<a NAME="E5:50_1_1_3"/>
<font color="Maroon" title="c7">x9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1_3"><span class="lab"><font color="Green" title="E44">A10</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:50_1_1_3"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
<font color="Maroon" title="c7">x9</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> 
 <span class="kw">and </span><br/><a NAME="E8:50_1_1_3"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
<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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1_3"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E7:50_1_1_3"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c8">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:50_1_1_3"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1_1_3"><span class="lab"><font color="Green" title="E46">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A13</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E9:50_1_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="50_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:50_1_1_4"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">q9</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><span class="kw"> such that </span><br/><a NAME="E3:50_1_1_4"/><span class="lab"><font color="Green" title="E44">A14</font></span>: 
<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">q9</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_1_4"/><span class="lab"><font color="Green" title="E45">A15</font></span>: 
<font color="Maroon" title="c7">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 ;<br/>
<span class="lab"><font color="Green" title="E46">A16</font></span>: <font color="Maroon" title="c7">q9</font> = 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c7">q9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c7">q9</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></span>
;<br/>
<a NAME="E6:50_1_1_4"/>
<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">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E4:50_1_1_4"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:50_1_1_4"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1_4"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E5:50_1_1_4"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:50_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></span>;<br/>
<a NAME="E11:50_1_1"/><span class="kw">then </span>
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:50_1_1"/>
 ex <font color="Olive" title="b1">D9</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">D9</font>,<font color="Maroon" title="c1">L</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1_1"><span class="lab"><font color="Green" title="E43">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:50_1"/>
<font color="Maroon" title="c2">D</font> is  <a href="lattice6.html#V9" title="LATTICE6:attr.9">infimum-dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice6.html#D14" target="_self" title="LATTICE6:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:50_2"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:50_2"/>
 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="50_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">consider </span><font color="Maroon" title="c4">D9</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E2:50_2_1"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
 <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">D9</font>,<font color="Maroon" title="c1">L</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="ref" href="lattice6.html#D14" target="_self" title="LATTICE6:def.14">Def14</a></span>;<br/>
<a NAME="E3:50_2_1"/><span class="lab"><font color="Green" title="E40">A19</font></span>: 
 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">D9</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="50_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">D9</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/>

<span class="kw">assume </span><a NAME="E1:50_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">D9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:50_2_1_1"/><span class="kw">then </span>
<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/>
<span class="kw">hence </span><a NAME="E3:50_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 <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:50_2_1"/><span class="lab"><font color="Green" title="E41">A20</font></span>: 
 <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">D9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_2_1"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">D9</font> = <font color="Maroon" title="c4">D9</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2_1"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E6:50_2_1"/><span class="lab"><font color="Green" title="E42">A21</font></span>: 
 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">D9</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="50_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">D9</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/>

<span class="lab"><font color="Green" title="E43">A22</font></span>: <font color="Maroon" title="c6">b</font> = 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c6">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E2:50_2_1_2"/><span class="lab"><font color="Green" title="E44">A23</font></span>: 
<font color="Maroon" title="c5">D9</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E3:50_2_1_2"/>
 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">D9</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="50_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">D9</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/>

<span class="kw">assume </span><a NAME="E1:50_2_1_2_2"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:50_2_1_2_2"/><span class="kw">then </span>
<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">D9</font> </span> } </span> 
 
;<br/>
<a NAME="E3:50_2_1_2_2"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_2_1_2"><span class="lab"><font color="Green" title="E44">A23</font></span></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:50_2_1_2_2"/>
 <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2_1_2"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:50_2_1_2"/><span class="kw">then </span>
 <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">D9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></span>;<br/>
<a NAME="E5:50_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A24</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_2_1"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></span>;<br/>
<font color="Maroon" title="c3">a</font> = 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:50_2_1_2"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2_1_2"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E5:50_2_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:50_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="c5">D9</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="50_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">D9</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/>

<span class="kw">assume </span><a NAME="E1:50_2_1_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D9</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:50_2_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x9</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">x9</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  &amp; <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D9</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:50_2_1_3"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:50_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A25</font></span>: 
<font color="Maroon" title="c5">D9</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E9:50_2_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">D9</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="50_2_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">D9</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/>

<span class="lab"><font color="Green" title="E44">A26</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E2:50_2_1_4"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D9</font> <a href="lattice6.html#K1" title="LATTICE6:func.1">%</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">u9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E4:50_2_1_4"/><span class="lab"><font color="Green" title="E45">A27</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u9</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 <span class="kw">and </span><br/><a NAME="E5:50_2_1_4"/><span class="lab"><font color="Green" title="E46">A28</font></span>: 
<font color="Maroon" title="c7">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D9</font>
 ;<br/>
<a NAME="E6:50_2_1_4"/>
 <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">u9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_2_1"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E5:50_2_1_4"><span class="lab"><font color="Green" title="E46">A28</font></span></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:50_2_1_4"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_2_1_4"><span class="lab"><font color="Green" title="E45">A27</font></span></a>, <a class="txt" href="#E1:50_2_1_4"><span class="lab"><font color="Green" title="E44">A26</font></span></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:50_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">D9</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></span>;<br/>
<a NAME="E11:50_2_1"/><span class="kw">then </span>
<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">D9</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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_2_1"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:50_2_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_2_1"><span class="lab"><font color="Green" title="E43">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:50_2"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="waybel_6.html#T15" title="WAYBEL_6:th.15">WAYBEL_6:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:50"/>
( <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
