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

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:43"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">R</font> is <a href="waybel_0.html#V16">connected</a>
 ;<br/>

<b>set </b><font color="Maroon">IR'</font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">CR'</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">IR</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">CR</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>let </b><font color="Maroon">N</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D10" target="_self">DICKSON:def 10</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1"/>
( <font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> or <font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>suppose </b></a><a NAME="E1:43_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">B</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><b>thus </b><a NAME="E2:43_1_1"/>
<font color="Maroon">B</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:43_1_1"><i><font color="Green">E30</font></i></a>, </i>;<br/><b>thus </b><a NAME="E3:43_1_1"/>
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2"><b>suppose </b></a><a NAME="E1:43_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:43_1_2"/><i><font color="Green">E32</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#R1">is_well_founded_in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="wellfnd1.html#D2">WELLFND1:def 2</a></i>;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:43_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E5:43_1_2"/><i><font color="Green">E34</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="txt" href="dickson.html#E1:43_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/><a NAME="E6:43_1_2"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="dickson.html#E4:43_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>take </b><font color="Maroon">B</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span>;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N</font> <b>by </b><i><a class="txt" href="dickson.html#E2:43_1_2"><i><font color="Green">E32</font></i></a></i>;<br/><b>thus </b><a NAME="E8:43_1_2"/>
<font color="Maroon">B</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_2_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T55">SUBSET_1:55</a></i>;<br/>
<b>hence </b><a NAME="E2:43_1_2_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D9" target="_self">DICKSON:def 9</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E3:43_1_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i><a class="txt" href="dickson.html#E2:43_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E5:43_1_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_2_1_1"/>
( <font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> or <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E2:43"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_2_1_1_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_2_1_1_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_2_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:43_1_2_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_2_1_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or  not <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_2_1_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_2_1_1_2_1_1_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_2_1_1_2_1_1_2"/><i><font color="Green">E71</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_2_1_1_2_1_1_2_1_1"/>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> or  not <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_2_1_1_2_1_1_2_1_1_1"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 ;<br/><div class="add"><a NAME="E2:43_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:43_1_2_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:43_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="dickson.html#E6:43_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_2_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E72</font></i>: 
not <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 ;<br/><div class="add"><a NAME="E2:43_1_2_1_1_2_1_1_2_1_1_2"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:43_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/><b>hence </b><a NAME="E4:43_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:43_1_2_1_1_2_1_1_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:43_1_2_1_1_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E9:43_1_2"/>
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
