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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice7.html#K3" title="LATTICE7:func.3">Join-IRR</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<a NAME="E2:20"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> &amp; ( for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">c</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c</font> ) ) )
 
<b>by </b><i><a class="ref" href="lattice7.html#T10" target="_self" title="LATTICE7:th.10">Th10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">z</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="E4:20"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="lattice7.html#R2" title="LATTICE7:pred.2">&lt;(1)</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="lattice7.html#T9" target="_self" title="LATTICE7:th.9">Th9</a></i>;<br/>
<a NAME="E5:20"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c3">z</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; ( for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Maroon" title="c3">z</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">v</font> or  not <font color="Olive" title="b1">v</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> ) ) )
 
<b>by </b><i><a class="txt" href="lattice7.html#E4:20"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="lattice7.html#D4" target="_self" title="LATTICE7:def.4">Def4</a></i>;<br/>
<a NAME="E6:20"/>
for <font color="Olive" title="b1">y</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">y</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> holds <br/><font color="Olive" title="b1">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E18">A4</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">z</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</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">g</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Olive" title="b1">g</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">z</font> ) </span>}</span> 
 ;<br/>
<a NAME="E4:20_1"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c5">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green" title="E21">A7</font></i>: 
not <font color="Maroon" title="c5">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<a NAME="E2:20_1_1"/><i><font color="Green" title="E22">A8</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> iff ( <font color="Olive" title="b1">t</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Olive" title="b1">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">z</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<a NAME="E1:20_1_1_1"/>
( <font color="Maroon" title="c6">t</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">t</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">z</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_1_1"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">g</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="E3:20_1_1_1_1"/><i><font color="Green" title="E23">A9</font></i>: 
( <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c7">g</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Maroon" title="c7">g</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">z</font> )
 <b>by </b><i><a class="txt" href="lattice7.html#E3:20_1"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:20_1_1_1_1"/>
( <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">z</font> )
 <b>by </b><i><a class="txt" href="lattice7.html#E3:20_1_1_1_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1_1"/>
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> iff ( <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Maroon" title="c6">t</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">z</font> ) )
 <b>by </b><i><a class="txt" href="lattice7.html#E3:20_1"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:20_1_1"/>
<font color="Maroon" title="c5">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_2"/>
<font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="20_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:20_1_1_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">g</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="E3:20_1_1_2_1"/><i><font color="Green" title="E23">A10</font></i>: 
( <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c7">g</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Maroon" title="c7">g</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">z</font> )
 <b>by </b><i><a class="txt" href="lattice7.html#E3:20_1"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:20_1_1_2_1"/>
<font color="Maroon" title="c6">f</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>
 <b>by </b><i><a class="txt" href="lattice7.html#E3:20_1_1_2_1"><i><font color="Green" title="E23">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1_2"/>
<font color="Maroon" title="c5">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>thus </b><a NAME="E3:20_1_1_2"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c6">a</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="E5:20_1_1"/><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>and </b><br/><a NAME="E6:20_1_1"/><i><font color="Green" title="E24">A12</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> holds <br/> not <font color="Maroon" title="c6">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">b</font>
 <b>by </b><i><a class="txt" href="lattice7.html#E1:20_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="lattice7.html#T8" target="_self" title="LATTICE7:th.8">Th8</a></i>;<br/>
<a NAME="E7:20_1_1"/><i><font color="Green" title="E25">A13</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Maroon" title="c6">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">z</font> )
 
<b>by </b><i><a class="txt" href="lattice7.html#E2:20_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="lattice7.html#E5:20_1_1"><i><font color="Green" title="E23">A11</font></i></a></i>;<br/>
<a NAME="E8:20_1_1"/><i><font color="Green" title="E26">A14</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_3"/>
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="lattice7.html#E2:20_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="lattice7.html#E5:20_1_1"><i><font color="Green" title="E23">A11</font></i></a></i>;<br/>
<a NAME="E2:20_1_1_3"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E3:20_1_1_3"/>
<font color="Maroon" title="c3">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="lattice7.html#E5:20"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E4:20_1_1_3"/><b>then </b><i><font color="Green" title="E28">A16</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="lattice7.html#E2:20_1_1_3"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="yellow_5.html#T9" title="YELLOW_5:th.9">YELLOW_5:9</a></i>;<br/>
<a NAME="E5:20_1_1_3"/>
<font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="lattice7.html#E1:20"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="lattice7.html#E5:20"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="lattice7.html#E7:20_1_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="lattice7.html#T10" target="_self" title="LATTICE7:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E6:20_1_1_3"/>
<font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="lattice7.html#E4:20_1_1_3"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:20_1_1"/>
not <font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">z</font>
 
<b>by </b><i><a class="txt" href="lattice7.html#E7:20_1_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="yellow_5.html#T3" title="YELLOW_5:th.3">YELLOW_5:3</a></i>;<br/>
<a NAME="E10:20_1_1"/><b>then </b><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="lattice7.html#E2:20_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="lattice7.html#E8:20_1_1"><i><font color="Green" title="E26">A14</font></i></a></i>;<br/>
<a NAME="E11:20_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font> <a href="orders_2.html#NR4" title="ORDERS_2:NR.4">&gt;</a> <font color="Maroon" title="c6">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_4"/>
<font color="Maroon" title="c6">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T5" title="YELLOW_5:th.5">YELLOW_5:5</a></i>;<br/>
<a NAME="E2:20_1_1_4"/><b>then </b><i><font color="Green" title="E28">A18</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>;<br/>
<a NAME="E3:20_1_1_4"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_4_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font>
 ;<br/>

<a NAME="E2:20_1_1_4_1"/>
<font color="Maroon" title="c3">z</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_4_1_1"/>
<font color="Maroon" title="c3">z</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T5" title="YELLOW_5:th.5">YELLOW_5:5</a></i>;<br/>
<a NAME="E2:20_1_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c3">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T2" title="YELLOW_5:th.2">YELLOW_5:2</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1_1_4_1_1"/>
<font color="Maroon" title="c3">z</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="lattice7.html#E7:20_1_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="lattice7.html#E2:20_1_1_4"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice7.html#E4:20"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="lattice7.html#E8:20_1_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="lattice7.html#D4" target="_self" title="LATTICE7:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:20_1_1_4"/>
<font color="Maroon" title="c6">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">z</font> <a href="orders_2.html#NR4" title="ORDERS_2:NR.4">&gt;</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="lattice7.html#E2:20_1_1_4"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:20_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice7.html#E6:20_1_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="lattice7.html#E10:20_1_1"><i><font color="Green" title="E27">A17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:20_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="lattice7.html#E1:20_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="lattice7.html#E3:20_1"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:20_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice7.html#E4:20_1"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:20"/>
 ex <font color="Olive" title="b1">z</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="Olive" title="b1">z</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; ( for <font color="Olive" title="b2">y</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="b2">y</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">x</font> holds <br/><font color="Olive" title="b2">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> ) )
 <b>by </b><i><a class="txt" href="lattice7.html#E5:20"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/>


</div>
