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

<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">R</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green">E29</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 <br/> (  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Olive">N</font> is <a href="finset_1.html#V1">finite</a> &amp; not  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Olive">N</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>

<a NAME="E3:50"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">R</font> is <a href="orders_2.html#V3">transitive</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:50"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> is <a href="orders_2.html#V2">reflexive</a>
 
<b>by </b><i>, </i>;<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>let </b><font color="Maroon">N</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D10" target="_self">DICKSON:def 10</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1"/>
( <font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> or  not <font color="Maroon">N</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>suppose </b></a><a NAME="E1:50_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">B</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><b>thus </b><a NAME="E2:50_1_1"/>
<font color="Maroon">B</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E3:50_1_1"/>
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><b>suppose </b></a><a NAME="E1:50_1_2"/>
not <font color="Maroon">N</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">N'</font> = <font color="Maroon">N</font> as  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>reconsider </b><font color="Maroon">MCN'</font> =  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">N'</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/><b>take </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">MCN'</font> : verum </span>}</span> ;<br/><b>thus </b><a NAME="E4:50_1_2"/>
<font color="Maroon">B</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">N</font>,<font color="Maroon">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_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/><b>assume </b><a NAME="E1:50_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">MCN'</font><b> such that </b><br/><a NAME="E3:50_1_2_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <font color="Maroon">y</font>
 ;<br/><b>consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E5:50_1_2_1_1"/>
<font color="Maroon">z</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="E6:50_1_2_1_1"/><i><font color="Green">E34</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">z</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/><a NAME="E7:50_1_2_1_1"/>
not <font color="Maroon">y</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E3:50_1_2_1"><i><font color="Green">E35</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:50_1_2_1_1"/>
<font color="Maroon">x</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_1_2_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <b>by </b><i><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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E3:50_1_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">NN'</font> = <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">N'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">d</font>] </span>}</span> ;<br/>
<a NAME="E4:50_1_2_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">N'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">d</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">N'</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E5:50_1_2_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/>
<a NAME="E6:50_1_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">N'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">d</font>] </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">NN'</font> = <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">N'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">d</font>] </span>}</span>  as  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> <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">NN'</font>;<br/>
<a NAME="E9:50_1_2_1"/><i><font color="Green">E69</font></i>: 
not  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">NN'</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="dickson.html#E3:50_1_2_1"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">m'</font>;<br/>
<b>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="E13:50_1_2_1"/>
<font color="Maroon">y</font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">NN'</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="E14:50_1_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">m'</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> <font color="Maroon">NN'</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E15:50_1_2_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c</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">y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NN'</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">c</font> = <font color="Maroon">c</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>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">N'</font><b> such that </b><br/><a NAME="E19:50_1_2_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>
<a NAME="E20:50_1_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c</font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">NN'</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="txt" href="dickson.html#E3:50_1_2_1"><i><font color="Green">E35</font></i></a>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_2_1_2"/>
not <font color="Maroon">c</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>
 ;<br/><b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_1_2_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E4:50_1_2_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E5:50_1_2_1_2"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">c</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>
 <b>by </b><i>, <a class="ref" href="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/><b>reconsider </b><font color="Maroon">w'</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:50_1_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">c</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">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/><a NAME="E8:50_1_2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">c</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">R</font>
 ;<br/><a NAME="E9:50_1_2_1_2"/><b>then </b>
<font color="Maroon">w'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E10:50_1_2_1_2"/><b>then </b>
<font color="Maroon">w'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E11:50_1_2_1_2"/><b>then </b>
<font color="Maroon">w'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NN'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E12:50_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, , , <a class="ref" href="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:50_1_2_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<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">c</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/>
<a NAME="E23:50_1_2_1"/><b>then </b><i><font color="Green">E91</font></i>: 
not <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">c</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="dickson.html#E3:50_1_2_1"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">t</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font></span>)</span>;<br/>
<a NAME="E24:50_1_2_1"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font></span>)</span> <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">t</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">c</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>thus </b><a NAME="E26:50_1_2_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>

<a NAME="E27:50_1_2_1"/>
<font color="Maroon">t</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">c</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E28:50_1_2_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">c</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="E29:50_1_2_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">c</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#E3:50_1_2_1"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E30:50_1_2_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">c</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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E31:50_1_2_1"/><b>then </b>
<font color="Maroon">t</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c'</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="E32:50_1_2_1"/>
<font color="Maroon">t</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a<sub>1</sub></font> =  <a href="subset_1.html#K10">choose</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/><a NAME="E5:50_1_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">MCN'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="pre_circ.html#S1">PRE_CIRC:sch 1</a>();<br/></i><b>hence </b><a NAME="E6:50_1_2"/>
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
