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

<b>let </b><font color="Maroon">L</font> be  <a href="group_1.html#V6">finite</a> <a href="waybel_1.html#V2">distributive</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="lattice7.html#K3" target="_self">Join-IRR</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:20"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> &amp; ( for <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  holds <br/>(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">c</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> ) ) )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:20"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="lattice7.html#R2" target="_self">&lt;(1)</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:20"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">z</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  holds <br/>(  not <font color="Maroon">z</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">v</font> or  not <font color="Olive">v</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> ) ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:20"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">y</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> holds <br/><font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">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">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/>

<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">g</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> &amp; not <font color="Olive">g</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> ) </span>}</span> 
 ;<br/>
<a NAME="E4:20_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Y</font> is <a href="xboole_0.html#V1">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">E49</font></i>: 
not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<a NAME="E2:20_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff ( <font color="Olive">t</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> &amp; not <font color="Olive">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">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">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<a NAME="E1:20_1_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies ( <font color="Maroon">t</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">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">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:20_1_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">g</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">g</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:20_1_1_1_1"/>
( <font color="Maroon">t</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff ( <font color="Maroon">t</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:20_1_1"/>
<font color="Maroon">Y</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">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">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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">f</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

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

<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:20_1_1_2_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">g</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:20_1_1_2_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1_2"/>
<font color="Maroon">Y</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">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">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:20_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E6:20_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> not <font color="Maroon">a</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">b</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:20_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:20_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">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">a</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:20_1_1_3"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E3:20_1_1_3"/>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E4:20_1_1_3"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="yellow_5.html#T9">YELLOW_5:9</a></i>;<br/>
<a NAME="E5:20_1_1_3"/>
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E6:20_1_1_3"/>
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:20_1_1"/>
not <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="yellow_5.html#T3">YELLOW_5:3</a></i>;<br/>
<a NAME="E10:20_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:20_1_1"/>
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font> <a href="orders_2.html#NR4" target="_self">&gt;</a> <font color="Maroon">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">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T5">YELLOW_5:5</a></i>;<br/>
<a NAME="E2:20_1_1_4"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>;<br/>
<a NAME="E3:20_1_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">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">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:20_1_1_4_1"/>
<font color="Maroon">z</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">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">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T5">YELLOW_5:5</a></i>;<br/>
<a NAME="E2:20_1_1_4_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T2">YELLOW_5:2</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1_1_4_1_1"/>
<font color="Maroon">z</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#D10">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>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:20_1_1_4"/>
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">z</font> <a href="orders_2.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D10">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>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:20_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:20_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:20"/>
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">y</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font> holds <br/><font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> ) )
 <b>by </b><i/>;<br/>


</div>
