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<b>let </b><font color="Maroon" title="c1">r</font>, <font color="Maroon" title="c2">s</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green" title="E29">A1</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">s</font>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c1">r</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 <b>and </b><br/><a NAME="E3:44"/><i><font color="Green" title="E31">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">s</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">s</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><br/>

<b>reconsider </b><font color="Maroon" title="c4">N'</font> = <font color="Maroon" title="c3">N</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">r</font> <b>by </b><i><a class="txt" href="dickson.html#E3:44"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:44"/><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c5">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c4">N'</font>,<font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c5">B</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#E2:44"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">B</font>
;<br/>

<b>thus </b><a NAME="E7:44"/>
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:44"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</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><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">s</font>;<br/><b>assume </b><a NAME="E1:44_1"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c6">a</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="dickson.html#E3:44"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">r</font><b> such that </b><br/><a NAME="E4:44_1"/><i><font color="Green" title="E34">A6</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c8">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">a'</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E6:44"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="dickson.html#E1:44_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">b'</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">s</font> <b>by </b><i><a class="txt" href="dickson.html#E3:44"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c9">b'</font>;<br/><a NAME="E6:44_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a'</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 <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:44_1"><i><font color="Green" title="E34">A6</font></i></a>, <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="E7:44_1"/>
( <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c10">b'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:44"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="dickson.html#E4:44_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E9:44"/>
<font color="Maroon" title="c5">B</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#E6:44"><i><font color="Green" title="E32">A4</font></i></a></i>;<br/>


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