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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</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/>



<a NAME="E1:11"/>
(  ex <font color="Olive">A</font> being   <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font> st  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">A</font> &amp; ( for <font color="Olive">A</font> being   <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font> holds   <a href="card_1.html#K4">card</a> <font color="Olive">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> ) )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font><b> such that </b><br/><a NAME="E3:11"/>
 <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:11"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">B</font> being   <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font> holds   <a href="card_1.html#K4">card</a> <font color="Olive">B</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font>
 ;<br/>
<a NAME="E5:11"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/>
<a NAME="E6:11"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">x</font></span><a href="domain_1.html#K7">}</a></span> is    <a href="lattice7.html#M1" target="_self">Chain</a> of  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>,<font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:11"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">x</font></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1"/>
( <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> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1"><b>suppose </b></a><a NAME="E1:11_1_1"/>
<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>
 ;<br/><div class="add"><a NAME="E2:11_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">x</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/><b>hence </b><a NAME="E3:11_1_1"/>
 <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2"><b>suppose </b></a><a NAME="E1:11_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><a NAME="E2:11_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="lattice7.html#K1" target="_self">height</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:11_1_2"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:11_1_2_1"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="lattice7.html#R1" target="_self">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice7.html#D1" target="_self">LATTICE7:def 1</a><br/>

<b>assume </b><a NAME="E1:11_1_2_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E2:11_1_2_1_1"/><b>then </b>
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> or <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:11_1_2_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:11_1_2_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="lattice7.html#R1" target="_self">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice7.html#D1" target="_self">LATTICE7:def 1</a><br/>

<b>assume </b><a NAME="E1:11_1_2_1_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:11_1_2_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:11_1_2_1_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_1_2_1"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:11_1_2"/>
 <a href="lattice7.html#K1" target="_self">height</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, , <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
