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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:52"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> </span>}</span> ;<br/>
<a NAME="E2:52"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 
<b>by </b><i><a class="ref" href="dickson.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:52"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="dickson.html#E1:52"><i><font color="Green">E36</font></i></a>, <a class="ref" href="dickson.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E4:52"/><b>then </b>
(  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; not  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 
<b>by </b><i><a class="txt" href="dickson.html#E2:52"><i><font color="Green">E37</font></i></a>, <a class="ref" href="dickson.html#T32" target="_self">Th32</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:52"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span><b> such that </b><br/><a NAME="E8:52"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E6:52"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E9:52"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E8:52"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<div><i><font color="Green">E40</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">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span><b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E4:52_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:52_1"><i><font color="Green">E41</font></i></a></i>;<br/><b>thus </b><a NAME="E5:52_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:52_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="dickson.html#E4:52_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:52"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><a NAME="E13:52"/>
<font color="Maroon">c<sub>9</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></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">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:52"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>; <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">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:52_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</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">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></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">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <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">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</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">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:52_2_1_1_1"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E3:52_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E2:52_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="dickson.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E4:52_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></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">E43</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">E44</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">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:52_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:52"><i><font color="Green">E36</font></i></a>, <a class="txt" href="dickson.html#E2:52"><i><font color="Green">E37</font></i></a>, <a class="ref" href="dickson.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E3:52_2_1_1_2"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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 <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="wellfnd1.html#D2">WELLFND1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:52_2_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <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">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E6:52_2_1_1_2"/><i><font color="Green">E46</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#R1">misses</a> <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">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:52_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/><a NAME="E7:52_2_1_1_2"/>
<font color="Maroon">c<sub>12</sub></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">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:52_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:52_2_1_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</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">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E9:52_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E12:52_2_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <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">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:52_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E6:52_2_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="dickson.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E13:52_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="dickson.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E14:52_2_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E15:52_2_1_1_2"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>1</sub></font> <a href="relset_1.html#K3">\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E8:52_2_1_1_2"><i><font color="Green">E47</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">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>1</sub></font>
 ;<br/><a NAME="E17:52_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">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">E43</font></i></a>, <a class="txt" href="dickson.html#E14:52_2_1_1_2"><i><font color="Green">E48</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">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E14:52"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E15:52"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E14:52"><i><font color="Green">E42</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">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">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:52_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:52"><i><font color="Green">E40</font></i></a>, <a class="txt" href="dickson.html#E1:52_3"><i><font color="Green">E44</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:52_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E12:52"><i><font color="Green">E41</font></i></a>, <a class="txt" href="dickson.html#E14:52"><i><font color="Green">E42</font></i></a>, <a class="txt" href="dickson.html#E1:52_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span><b> such that </b><br/><a NAME="E7:52_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:52_3"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E8:52_3"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:52_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E9:52_3"/><i><font color="Green">E48</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D6">ORDERS_2:def 6</a></i>;<br/><a NAME="E10:52_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E15:52"><i><font color="Green">E43</font></i></a>, <a class="txt" href="dickson.html#E5:52_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E7:52_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="dickson.html#E8:52_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="dickson.html#T18" target="_self">Th18</a></i>;<br/><b>hence </b><a NAME="E11:52_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:52_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E8:52_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="dickson.html#E9:52_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:52"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div>
