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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a> &amp; ( for <font color="Olive" title="b1">N</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> holds <br/> (  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Olive" title="b1">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; not  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Olive" title="b1">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) ) implies <font color="Maroon" title="c1">R</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green" title="E36">A2</font></i>: 
for <font color="Olive" title="b1">N</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> holds <br/> (  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Olive" title="b1">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; not  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Olive" title="b1">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">R</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a></span><br/>

<a NAME="E3:50"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 
<b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></i>;<br/>
<a NAME="E4:50"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>
 
<b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c2">IR</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c3">CR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">IR'</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c5">N</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">DICKSON:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Olive" title="b1">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><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" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> or  not <font color="Maroon" title="c5">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="E39">A5</font></i>: 
<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Olive" title="b1">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/><div class="add"><b>take </b><font color="Maroon" title="c6">B</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c6">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/><b>thus </b><a NAME="E2:50_1_1"/>
<font color="Maroon" title="c6">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="#E1:50_1_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="dickson.html#T26" target="_self" title="DICKSON:th.26">Th26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/><b>thus </b><a NAME="E3:50_1_1"/>
<font color="Maroon" title="c6">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c5">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Olive" title="b1">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">N'</font> = <font color="Maroon" title="c5">N</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>reconsider </b><font color="Maroon" title="c7">MCN'</font> =  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c6">N'</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E36">A2</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c8">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">MCN'</font> : verum </span>}</span> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c8">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/><b>thus </b><a NAME="E4:50_1_2"/>
<font color="Maroon" title="c8">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c5">N</font>,<font color="Maroon" title="c1">R</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><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" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> )</span><br/><b>assume </b><a NAME="E1:50_1_2_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font></span><br/><b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">MCN'</font><b> such that </b><br/><a NAME="E3:50_1_2_1_1"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <font color="Maroon" title="c10">y</font>
 ;<br/><b>consider </b><font color="Maroon" title="c11">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span><b> such that </b><br/><a NAME="E5:50_1_2_1_1"/>
<font color="Maroon" title="c11">z</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c5">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>and </b><br/><a NAME="E6:50_1_2_1_1"/><i><font color="Green" title="E40">A7</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c11">z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="ref" href="dickson.html#D8" target="_self" title="DICKSON:def.8">Def8</a></i>;<br/><a NAME="E7:50_1_2_1_1"/>
not <font color="Maroon" title="c10">y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="dickson.html#T23" target="_self" title="DICKSON:th.23">Th23</a></i>;<br/><b>hence </b><a NAME="E8:50_1_2_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="txt" href="#E3:50_1_2_1_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="#E6:50_1_2_1_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_1_2_1"/>
<font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">DICKSON:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> holds <br/> ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> )</span><br/>

<b>let </b><font color="Maroon" title="c9">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> implies  ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Olive" title="b1">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font> ) )</span><br/>

<b>assume </b><a NAME="E3:50_1_2_1"/><i><font color="Green" title="E39">A8</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Olive" title="b1">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font> )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>] <b>means </b>$1 <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font>;<br/>
<b>set </b><font color="Maroon" title="c10">NN'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">N'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>] </span>}</span> ;<br/>
<a NAME="E4:50_1_2_1"/><i><font color="Green" title="E40">A9</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">N'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>] </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c6">N'</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E5:50_1_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font>
 
<b>by </b><i><a class="txt" href="#E4:50"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></i>;<br/>
<a NAME="E6:50_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">N'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E3:50_1_2_1"><i><font color="Green" title="E39">A8</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">NN'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">N'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>] </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="#E4:50_1_2_1"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c11">NN'</font>;<br/>
<a NAME="E9:50_1_2_1"/><i><font color="Green" title="E41">A10</font></i>: 
not  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c11">NN'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E36">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">m'</font> = <font color="Maroon" title="c12">m</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="dickson.html#T23" target="_self" title="DICKSON:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">m'</font>;<br/>
<b>consider </b><font color="Maroon" title="c15">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span><b> such that </b><br/><a NAME="E13:50_1_2_1"/>
<font color="Maroon" title="c15">y</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c11">NN'</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>and </b><br/><a NAME="E14:50_1_2_1"/><i><font color="Green" title="E42">A11</font></i>: 
<font color="Maroon" title="c13">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c15">y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">NN'</font>
 <b>by </b><i><a class="txt" href="#E9:50_1_2_1"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="dickson.html#D8" target="_self" title="DICKSON:def.8">Def8</a></i>;<br/>
<a NAME="E15:50_1_2_1"/><i><font color="Green" title="E43">A12</font></i>: 
( <font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c15">y</font> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">NN'</font> )
 
<b>by </b><i><a class="txt" href="#E14:50_1_2_1"><i><font color="Green" title="E42">A11</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">c</font> = <font color="Maroon" title="c14">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">c'</font> = <font color="Maroon" title="c16">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>consider </b><font color="Maroon" title="c18">d</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">N'</font><b> such that </b><br/><a NAME="E19:50_1_2_1"/><i><font color="Green" title="E44">A13</font></i>: 
( <font color="Maroon" title="c16">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">d</font> &amp; <font color="Maroon" title="c18">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font> )
 <b>by </b><i><a class="txt" href="#E15:50_1_2_1"><i><font color="Green" title="E43">A12</font></i></a></i>;<br/>
<a NAME="E20:50_1_2_1"/><i><font color="Green" title="E45">A14</font></i>: 
<font color="Maroon" title="c16">c</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c11">NN'</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E9:50_1_2_1"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="dickson.html#T20" target="_self" title="DICKSON:th.20">Th20</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" title="c16">c</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c5">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c19">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_1_2_1_2"/><i><font color="Green" title="E46">A15</font></i>: 
<font color="Maroon" title="c19">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <b>and </b><br/><a NAME="E4:50_1_2_1_2"/><i><font color="Green" title="E47">A16</font></i>: 
<font color="Maroon" title="c19">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">c</font>
 <b>and </b><br/><a NAME="E5:50_1_2_1_2"/><i><font color="Green" title="E48">A17</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">w</font>,<font color="Maroon" title="c16">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E19:50_1_2_1"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="waybel_4.html#D26" title="WAYBEL_4:def.26">WAYBEL_4:def 26</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">w'</font> = <font color="Maroon" title="c19">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="#E5:50_1_2_1_2"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:50_1_2_1_2"/>
<font color="Maroon" title="c20">w'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c17">c'</font>
 <b>by </b><i><a class="txt" href="#E5:50_1_2_1_2"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E8:50_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c20">w'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="#E3:50"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E19:50_1_2_1"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><a NAME="E9:50_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c20">w'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">NN'</font>
 <b>by </b><i><a class="txt" href="#E3:50_1_2_1_2"><i><font color="Green" title="E46">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E10:50_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E20:50_1_2_1"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="#E4:50_1_2_1_2"><i><font color="Green" title="E47">A16</font></i></a>, <a class="txt" href="#E5:50_1_2_1_2"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="waybel_4.html#D26" title="WAYBEL_4:def.26">WAYBEL_4:def 26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E22:50_1_2_1"/><b>then </b><i><font color="Green" title="E46">A18</font></i>: 
<span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c16">c</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c5">N</font>
 
<b>by </b><i><a class="ref" href="dickson.html#D8" target="_self" title="DICKSON:def.8">Def8</a></i>;<br/>
<a NAME="E23:50_1_2_1"/><b>then </b><i><font color="Green" title="E47">A19</font></i>: 
not <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c16">c</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="dickson.html#T23" target="_self" title="DICKSON:th.23">Th23</a></i>;<br/>
<b>set </b><font color="Maroon" title="c19">t</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c16">c</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span>;<br/>
<a NAME="E24:50_1_2_1"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c16">c</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 
<b>by </b><i><a class="txt" href="#E23:50_1_2_1"><i><font color="Green" title="E47">A19</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">t</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c16">c</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c20">t</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c20">t</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font> )</span><br/>

<b>thus </b><a NAME="E26:50_1_2_1"/>
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="txt" href="#E22:50_1_2_1"><i><font color="Green" title="E46">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">t</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font></span><br/>

<a NAME="E27:50_1_2_1"/>
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c16">c</font>
 
<b>by </b><i><a class="txt" href="#E23:50_1_2_1"><i><font color="Green" title="E47">A19</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E28:50_1_2_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c20">t</font>,<font color="Maroon" title="c16">c</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">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" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c20">t</font>,<font color="Maroon" title="c16">c</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="dickson.html#D4" target="_self" title="DICKSON:def.4">Def4</a></i>;<br/>
<a NAME="E30:50_1_2_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c20">t</font>,<font color="Maroon" title="c16">c</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E31:50_1_2_1"/><b>then </b>
<font color="Maroon" title="c20">t</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c17">c'</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E32:50_1_2_1"/>
<font color="Maroon" title="c20">t</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="#E3:50"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E19:50_1_2_1"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of $1 =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> $1;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">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" title="b1">x</font>) where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">MCN'</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>from </b><i><a class="ref" href="pre_circ.html#S1" title="PRE_CIRC:sch.1">PRE_CIRC:sch 1</a>();<br/></i><b>hence </b><a NAME="E6:50_1_2"/>
<font color="Maroon" title="c8">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
