<?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><a NAME="E1:37"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">R</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">R</font> is <a href="waybel_0.html#V16">connected</a> &amp; <font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a>  is <a href="wellfnd1.html#V1">well_founded</a> )
 ;<br/><b>let </b><font color="Maroon">N</font> be  non <a href="xboole_0.html#V1">empty</a> <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>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1"/><i><font color="Green">E30</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="E5:37_1"/><i><font color="Green">E31</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="E6:37_1"/><i><font color="Green">E32</font></i>: 
<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>, </i>;<br/><a NAME="E7:37_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; ( for <font color="Olive">b</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>  holds <br/>(  not <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> or  not <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> or  not <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Olive">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 <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="txt" href="dickson.html#E1:37"><i><font color="Green">E26</font></i></a>, <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="37_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">N</font>
 ;<br/><b>then consider </b><font color="Maroon">b</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="E3:37_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">b</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="E4:37_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">y</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">b</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/><a NAME="E5:37_1_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; ( for <font color="Olive">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>  holds <br/>(  not <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> or  not <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> or  not <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Olive">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> ) ) )
 <b>by </b><i><a class="txt" href="dickson.html#E1:37_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</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> ;<br/><div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_1"/>
( <font color="Maroon">a'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b'</font> or <font color="Maroon">b'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a'</font> )
 </span><b>by </b><i>, <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="37_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1"/><i><font color="Green">E70</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="E2:37_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E71</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="37_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1_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="E2:37_1_1_1_1_1_1_1_1_1"/>
<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="txt" href="dickson.html#E7:37_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_1_1_1_1_1_1_1_2_1"/>
not <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>
 ;<br/><a NAME="E2:37_1_1_1_1_1_1_1_1_2_1"/><b>then </b>
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 <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/><a NAME="E3:37_1_1_1_1_1_1_1_1_2_1"/><b>then </b>
<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#K3">\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:37_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:37_1_1_1_1_1_1_1_1_2_1"/><b>then </b>
<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/><b>hence </b><a NAME="E5:37_1_1_1_1_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="dickson.html#E1:37_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="dickson.html#E4:37_1_1_1"><i><font color="Green">E35</font></i></a>, <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="E3:37_1_1_1_1_1_1_1_1_2"/>
<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>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:37_1_1_1_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 <font color="Maroon">R</font> &amp; <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="txt" href="dickson.html#E7:37_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">b'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a'</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<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="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_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="E1:37_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_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="E2:37_1_1_1_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 <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:37_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_1_1_1_1_2_1_1_2_1"/>
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 <font color="Maroon">R</font>
 ;<br/><a NAME="E2:37_1_1_1_1_1_2_1_1_2_1"/><b>then </b>
not <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> <a href="relset_1.html#K6">~</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/><a NAME="E3:37_1_1_1_1_1_2_1_1_2_1"/><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> <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">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:37_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:37_1_1_1_1_1_2_1_1_2_1"/><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 <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 ;<br/><b>hence </b><a NAME="E5:37_1_1_1_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:37"><i><font color="Green">E26</font></i></a>, <a class="txt" href="dickson.html#E5:37_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="dickson.html#E4:37_1_1_1"><i><font color="Green">E35</font></i></a>, <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="E3:37_1_1_1_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>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:37_1_1_1_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> &amp; <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="txt" href="dickson.html#E7:37_1"><i><font color="Green">E33</font></i></a>, <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><a NAME="E9:37_1_1_1"/><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> <a href="relset_1.html#K6">~</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/><a NAME="E10:37_1_1_1"/><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> <a href="eqrel_1.html#K2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E6:37_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E11:37_1_1_1"/><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>  <a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="dickson.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E12:37_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E13:37_1_1_1"/><b>then </b>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/><b>hence </b><a NAME="E14:37_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="txt" href="dickson.html#E3:37_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:37_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E3:37_1_1"/><b>then </b>
<font color="Maroon">y</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1"/>
<font color="Maroon">y</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><a class="txt" href="dickson.html#E1:37"><i><font color="Green">E26</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E9:37_1"/><b>then </b>
 <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E10:37_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:37"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">N</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> holds   <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="dickson.html#K6" target="_self">min-classes</a> <font color="Olive">N</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:37_2"/><i><font color="Green">E90</font></i>: 
( not <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; not <font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> )
 ;<br/><a NAME="E2:37_2"/><b>then </b><i><font color="Green">E91</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 <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E3:37_2"/><b>then </b><i><font color="Green">E92</font></i>: 
not <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> <a href="relset_1.html#K6">~</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/><a NAME="E4:37_2"/><i><font color="Green">E93</font></i>: 
not <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="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E5:37_2"/><b>then </b><i><font color="Green">E94</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 <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/><b>set </b><font color="Maroon">N'</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>;<br/><b>set </b><font color="Maroon">MCN</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="domain_1.html#K7">}</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K6" target="_self">min-classes</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><b>then consider </b><font color="Maroon">y</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="E3:37_2_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">y</font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>,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="E4:37_2_1_1"/><i><font color="Green">E96</font></i>: 
<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">y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:37_2_1_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> &amp; ( for <font color="Olive">z</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>  holds <br/>(  not <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> 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="yellow_3.html#K7">[</a><span class="default"><font color="Olive">z</font>,<font color="Maroon">y</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="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_2_1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:37_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:37_2_1_1_1_1_1_1_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>  <a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E3:37_2_1_1_1_1_1_1_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> <a href="eqrel_1.html#K2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="dickson.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E4:37_2_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:37_2_1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:37_2_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E3:37_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:37_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:37_2_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/><b>hence </b><a NAME="E6:37_2_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E3:37_2_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E4:37_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">y</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="37_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_2_1_1_1_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:37_2_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_1_1_2_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_2_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:37_2_1_1_1_2_1_1_1_1_2"/><b>then </b>
<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>  <a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E3:37_2_1_1_1_2_1_1_1_1_2"/><b>then </b>
<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="eqrel_1.html#K2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="dickson.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E4:37_2_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:37"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:37_2_1_1_1_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:37_2_1_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E3:37_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:37_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:37_2_1_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/><b>hence </b><a NAME="E6:37_2_1_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E3:37_2_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E4:37_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:37_2_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2_1"><b>suppose </b></a><a NAME="E1:37_2_1_2_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_2_1_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</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> ;<br/><b>take </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font>;<br/><a NAME="E2:37_2_1_2_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2_1_2_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">a'</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">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_2_1_2_1_1_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E4:37_2_1_2_1_1_1"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a'</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="E5:37_2_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:37_2_1_2_1_1_1"/><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> <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">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:37_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:37_2_1_2_1_1"/>
<font color="Maroon">a'</font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>,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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_2_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_2_1_2_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:37_2_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:37_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/><b>hence </b><a NAME="E4:37_2_1_2_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <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>assume </b><a NAME="E2:37_2_1_2_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E3:37_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">a</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2_1_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_2_1_1_2_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2_1_1_2_2_1"><b>suppose </b></a><a NAME="E1:37_2_1_2_1_1_2_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_2_1_2_1_1_2_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2_1_1_2_2_2"><b>suppose </b></a><a NAME="E1:37_2_1_2_1_1_2_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:37_2_1_2_1_1_2_2_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>  <a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E3:37_2_1_2_1_1_2_2_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> <a href="eqrel_1.html#K2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="dickson.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E4:37_2_1_2_1_1_2_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:37_2_1_2_1_1"/>
<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> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:37_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K6" target="_self">min-classes</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2_2"><b>suppose </b></a><a NAME="E1:37_2_1_2_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_2_2_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</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> ;<br/><b>take </b><font color="Maroon">b'</font> = <font color="Maroon">b'</font>;<br/><a NAME="E2:37_2_1_2_2_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2_1_2_2_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">b'</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">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_2_1_2_2_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E4:37_2_1_2_2_1_1"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">b'</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="E5:37_2_1_2_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:37_2_1_2_2_1_1"/><b>then </b>
<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#K3">\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:37_2_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:37"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:37_2_1_2_2_1"/>
<font color="Maroon">b'</font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>,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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_2_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_2_1_2_2_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:37_2_1_2_2_1_2_1"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:37_2_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/><b>hence </b><a NAME="E4:37_2_1_2_2_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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">b</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <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>assume </b><a NAME="E2:37_2_1_2_2_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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">b</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E3:37_2_1_2_2_1_2"/><b>then </b><i><font color="Green">E140</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">b</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2_2_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_2_2_1_2_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2_2_1_2_2_1"><b>suppose </b></a><a NAME="E1:37_2_1_2_2_1_2_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_2_1_2_2_1_2_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2_2_1_2_2_2"><b>suppose </b></a><a NAME="E1:37_2_1_2_2_1_2_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:37_2_1_2_2_1_2_2_2"/><b>then </b>
<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>  <a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E3:37_2_1_2_2_1_2_2_2"/><b>then </b>
<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="eqrel_1.html#K2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="dickson.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E4:37_2_1_2_2_1_2_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:37_2_1_2_2_1"/>
<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">b'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:37_2_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K6" target="_self">min-classes</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:37_2"/><b>then </b>
 <a href="dickson.html#K6" target="_self">min-classes</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E8:37_2"/><b>then </b><i><font color="Green">E141</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:37_2_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/><a NAME="E3:37_2_2"/>
<font color="Maroon">R</font> is <a href="orders_2.html#V2">reflexive</a>
 <b>by </b><i><a class="ref" href="dickson.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E4:37_2_2"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D4">ORDERS_2:def 4</a></i>;<br/><b>hence </b><a NAME="E5:37_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:37"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:37_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:37"/>
<font color="Maroon">R</font> is <a href="waybel_0.html#V16">connected</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:37_3"/><i><font color="Green">E143</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/><a NAME="E2:37_3"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E3:37_3"/><b>then </b>
 <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/><b>hence </b><a NAME="E4:37_3"/>
 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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:37"/>
<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/>;<br/>


</div>
