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<b>set </b><font color="Maroon" title="c3">BB</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">N</font> : <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="c2">N</font>,<font color="Maroon" title="c1">R</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c4">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>consider </b><font color="Maroon" title="c5">bp</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:53_1_1"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c5">bp</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E3:53_1_1"/>
<font color="Maroon" title="c5">bp</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:53"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></i>;<br/>
<a NAME="E4:53_1_1"/>
<font color="Maroon" title="c5">bp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E2:53_1_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<a NAME="E5:53_1_1"/><b>then </b><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c5">bp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">N</font> : <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="c2">N</font>,<font color="Maroon" title="c1">R</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E2:53_1_1"><i><font color="Green" title="E37">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">N</font> : <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="c2">N</font>,<font color="Maroon" title="c1">R</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c7">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E3:53_1_1_1"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 <b>and </b><br/><a NAME="E4:53_1_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E5:53_1_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:53_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> 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="dickson.html#E3:53_1_1_1"><i><font color="Green" title="E39">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:53_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">N</font> : <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="c2">N</font>,<font color="Maroon" title="c1">R</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> 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="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">BB</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">N</font> : <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="c2">N</font>,<font color="Maroon" title="c1">R</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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="dickson.html#E5:53_1_1"><i><font color="Green" title="E38">A3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">BB</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_1_1_2"/>
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">BB</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">b</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E3:53_1_1_2"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font>
 <b>and </b><br/><a NAME="E4:53_1_1_2"/><i><font color="Green" title="E40">A6</font></i>: 
<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="c2">N</font>,<font color="Maroon" title="c1">R</font>
 ;<br/><b>thus </b><a NAME="E5:53_1_1_2"/>
<font color="Maroon" title="c7">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:53_1_1_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="dickson.html#E4:53_1_1_2"><i><font color="Green" title="E40">A6</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E10:53_1_1"/><i><font color="Green" title="E39">A7</font></i>: 
<font color="Maroon" title="c7">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">R</font>
 ;<br/>

<a NAME="E11:53_1_1"/><b>then </b>
<font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">N</font>
 
<b>by </b><i><a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E12:53_1_1"/>
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">BB</font>
 <b>by </b><i><a class="txt" href="dickson.html#E10:53_1_1"><i><font color="Green" title="E39">A7</font></i></a></i>;<br/>


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