<?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>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>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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 ;<br/><a NAME="E2:34_1"/><b>then </b><i><font color="Green">E26</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>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>;<br/><b>assume </b><a NAME="E3:34_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>reconsider </b><font color="Maroon">N'</font> = <font color="Maroon">N</font> as   <a href="subset_1.html#NM2">Subset</a> of 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>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:34_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font>
 <b>and </b><br/><a NAME="E7:34_1"/><i><font color="Green">E31</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">y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">N'</font>
 <b>by </b><i>, , <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/><a NAME="E8:34_1"/><i><font color="Green">E32</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">y</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="ref" href="dickson.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</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/>;<br/><b>set </b><font color="Maroon">x</font> = <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">R</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:34_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_1_1"/>
 ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z</font>,<font color="Maroon">y</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">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> )
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>and </b><br/><a NAME="E4:34_1_1_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</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">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 ;<br/><a NAME="E5:34_1_1_1"/>
<font color="Maroon">z</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">y</font>
 <b>by </b><i>, <a class="txt" href="dickson.html#E2:34_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:34_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:34_1_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> or  not <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> or  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z</font>,<font color="Maroon">y</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">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> )
 ;<br/></div><b>end;</b></div><a NAME="E11:34_1"/><b>then </b>
<font color="Maroon">y</font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">N</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>
 <b>by </b><i><a class="ref" href="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/><a NAME="E12:34_1"/><b>then </b>
<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">R</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:34_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">N</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:34"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">N</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>  st <font color="Olive">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Olive">N</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">N</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:34_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</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>and </b><br/><a NAME="E2:34_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>reconsider </b><font color="Maroon">N'</font> = <font color="Maroon">N</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:34_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">N'</font>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span><b> such that </b><br/><a NAME="E7:34_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">a</font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">N'</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>
 <b>and </b><br/><a NAME="E8:34_2"/>
<font color="Maroon">x</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">R</font></span>)</span>,<font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N'</font>
 <b>by </b><i><a class="ref" href="dickson.html#T1" target="_self">Th1</a>, </i>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font>;<br/><b>thus </b><a NAME="E10:34_2"/>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_2_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">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a'</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_2_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">z</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">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a'</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:34_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">z</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">a'</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</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/>;<br/><a NAME="E6:34_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">z</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 <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="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><b>hence </b><a NAME="E7:34_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:34_2"/>
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">a'</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:34"/><b>then </b>
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="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:34"/>
<font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#D2">WELLFND1:def 2</a></i>;<br/>


</div>
