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

<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">N</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">IR</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">CR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c5">IR'</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> : <font color="Olive" title="b1">b</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> </span>}</span> ;<br/>
<a NAME="E2:52"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>
 
<b>by </b><i><a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></i>;<br/>
<a NAME="E3:52"/><b>then </b>
for <font color="Olive" title="b1">f</font> being  <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">R</font>  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b1">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">j</font> )
 
<b>by </b><i><a class="txt" href="dickson.html#E1:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="dickson.html#T30" target="_self" title="DICKSON:th.30">Th30</a></i>;<br/>
<a NAME="E4:52"/><b>then </b>
(  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c2">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; not  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c2">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="txt" href="dickson.html#E2:52"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="dickson.html#T32" target="_self" title="DICKSON:th.32">Th32</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:52"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span><b> such that </b><br/><a NAME="E8:52"/><i><font color="Green" title="E39">A4</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c8">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">N</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E6:52"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="dickson.html#D8" target="_self" title="DICKSON:def.8">Def8</a></i>;<br/>
<a NAME="E9:52"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> : <font color="Olive" title="b1">b</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E8:52"><i><font color="Green" title="E39">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> : <font color="Olive" title="b1">b</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c9">B</font>
;<br/>

<div><i><font color="Green" title="E40">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1"/><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/><b>consider </b><font color="Maroon" title="c11">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span><b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font>
 <b>and </b><br/><a NAME="E4:52_1"/><i><font color="Green" title="E43">A8</font></i>: 
<font color="Maroon" title="c11">b</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:52_1"><i><font color="Green" title="E41">A6</font></i></a></i>;<br/><b>thus </b><a NAME="E5:52_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:52_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="dickson.html#E4:52_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="waybel_4.html#D26" title="WAYBEL_4:def.26">WAYBEL_4:def 26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:52"/><b>then </b><i><font color="Green" title="E41">A9</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">N</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><a NAME="E13:52"/>
<font color="Maroon" title="c9">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_2"/>
<font color="Maroon" title="c9">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:52"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">DICKSON:def 9</a><br/>

<b>let </b><font color="Maroon" title="c10">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>

<b>assume </b><a NAME="E2:52_2"/><i><font color="Green" title="E42">A10</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">N</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">a'</font> = <font color="Maroon" title="c10">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2_1"/><i><font color="Green" title="E43">A11</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">R</font>  holds <br/>(  not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> or  not <font color="Olive" title="b1">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">a</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_2_1_1"/>
( <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_2_1_1_1"/>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:52_2_1_1_1"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E3:52_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">a'</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E2:52_2"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="dickson.html#T7" target="_self" title="DICKSON:th.7">Th7</a></i>;<br/><a NAME="E4:52_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/><b>hence </b><a NAME="E5:52_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:52_2_1"><i><font color="Green" title="E43">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_2_1_1_2"/><i><font color="Green" title="E44">A12</font></i>: 
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:52_2_1_1_2"/>
<font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="dickson.html#E2:52"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="dickson.html#T34" target="_self" title="DICKSON:th.34">Th34</a></i>;<br/><a NAME="E3:52_2_1_1_2"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="wellfnd1.html#D2" title="WELLFND1:def.2">WELLFND1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:52_2_1_1_2"/><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">N</font>
 <b>and </b><br/><a NAME="E6:52_2_1_1_2"/><i><font color="Green" title="E46">A14</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">z</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:52_2_1_1_2"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a></i>;<br/><a NAME="E7:52_2_1_1_2"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:52_2_1_1_2"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:52_2_1_1_2"/><b>then </b><i><font color="Green" title="E47">A15</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z</font>,<font color="Maroon" title="c10">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/><a NAME="E9:52_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">z</font> = <font color="Maroon" title="c12">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c14">z'</font> = <font color="Maroon" title="c13">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/><a NAME="E12:52_2_1_1_2"/>
<font color="Maroon" title="c13">z</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">a'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:52_2_1_1_2"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="dickson.html#E6:52_2_1_1_2"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="dickson.html#T7" target="_self" title="DICKSON:th.7">Th7</a></i>;<br/><a NAME="E13:52_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">z</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="dickson.html#T8" target="_self" title="DICKSON:th.8">Th8</a></i>;<br/><a NAME="E14:52_2_1_1_2"/><b>then </b><i><font color="Green" title="E48">A16</font></i>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/><a NAME="E15:52_2_1_1_2"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c10">a</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E8:52_2_1_1_2"><i><font color="Green" title="E47">A15</font></i></a></i>;<br/><a NAME="E16:52_2_1_1_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c10">a</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E17:52_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>hence </b><a NAME="E18:52_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:52_2_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="txt" href="dickson.html#E14:52_2_1_1_2"><i><font color="Green" title="E48">A16</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:52_2"/>
 ex <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">R</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> &amp; <font color="Olive" title="b1">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">a</font> )
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c10">C</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E14:52"/><i><font color="Green" title="E42">A17</font></i>: 
<font color="Maroon" title="c10">C</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font>
 ;<br/>

<a NAME="E15:52"/><i><font color="Green" title="E43">A18</font></i>: 
<font color="Maroon" title="c10">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E14:52"><i><font color="Green" title="E42">A17</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:52_3"/><i><font color="Green" title="E44">A19</font></i>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 ;<br/><a NAME="E2:52_3"/>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:52"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="dickson.html#E1:52_3"><i><font color="Green" title="E44">A19</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">b'</font> = <font color="Maroon" title="c11">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>consider </b><font color="Maroon" title="c13">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E5:52_3"/><i><font color="Green" title="E45">A20</font></i>: 
( <font color="Maroon" title="c13">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font> &amp; <font color="Maroon" title="c13">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">b'</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E12:52"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="dickson.html#E14:52"><i><font color="Green" title="E42">A17</font></i></a>, <a class="txt" href="dickson.html#E1:52_3"><i><font color="Green" title="E44">A19</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">b''</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span><b> such that </b><br/><a NAME="E7:52_3"/><i><font color="Green" title="E46">A21</font></i>: 
( <font color="Maroon" title="c14">b''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> &amp; <font color="Maroon" title="c14">b''</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c2">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:52_3"><i><font color="Green" title="E44">A19</font></i></a></i>;<br/><a NAME="E8:52_3"/><i><font color="Green" title="E47">A22</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">c</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:52_3"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E9:52_3"/><i><font color="Green" title="E48">A23</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D6" title="ORDERS_2:def.6">ORDERS_2:def 6</a></i>;<br/><a NAME="E10:52_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c13">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E15:52"><i><font color="Green" title="E43">A18</font></i></a>, <a class="txt" href="dickson.html#E5:52_3"><i><font color="Green" title="E45">A20</font></i></a>, <a class="txt" href="dickson.html#E7:52_3"><i><font color="Green" title="E46">A21</font></i></a>, <a class="txt" href="dickson.html#E8:52_3"><i><font color="Green" title="E47">A22</font></i></a>, <a class="ref" href="dickson.html#T18" target="_self" title="DICKSON:th.18">Th18</a></i>;<br/><b>hence </b><a NAME="E11:52_3"/>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:52_3"><i><font color="Green" title="E45">A20</font></i></a>, <a class="txt" href="dickson.html#E8:52_3"><i><font color="Green" title="E47">A22</font></i></a>, <a class="txt" href="dickson.html#E9:52_3"><i><font color="Green" title="E48">A23</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:52"/>
<font color="Maroon" title="c9">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div>
