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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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47"/>
<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="E2:47"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">f</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font>;<br/>

<b>set </b><font color="Maroon" title="c3">N</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>;<br/>
<a NAME="E3:47"/><i><font color="Green" title="E33">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:47"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E6:47"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c4">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#E2:47"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self" title="DICKSON:def.10">Def10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c4">B</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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="dickson.html#E5:47"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="dickson.html#E6:47"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="dickson.html#T27" target="_self" title="DICKSON:th.27">Th27</a></i>;<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/>
<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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <font color="Maroon" title="c2">f</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> $1;<br/>
<b>set </b><font color="Maroon" title="c6">BI</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] </span>}</span> ;<br/>
<a NAME="E8:47"/><i><font color="Green" title="E36">A5</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</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>consider </b><font color="Maroon" title="c7">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font>;<br/>
<a NAME="E10:47"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] </span>}</span> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c9">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font><b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Maroon" title="c9">b</font>
 ;<br/><b>thus </b><a NAME="E4:47_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:47_1"><i><font color="Green" title="E38">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">BI</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] </span>}</span>  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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="dickson.html#E8:47"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="dickson.html#E10:47"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">mB</font> =  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c8">BI</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c9">mB</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon" title="c11">fj</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">mB</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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/>
<a NAME="E15:47"/>
<font color="Maroon" title="c11">fj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E3:47"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">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="E17:47"/><i><font color="Green" title="E38">A8</font></i>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>and </b><br/><a NAME="E18:47"/><i><font color="Green" title="E39">A9</font></i>: 
<font color="Maroon" title="c12">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">fj</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:47"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<a NAME="E19:47"/><i><font color="Green" title="E40">A10</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E5:47"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self" title="DICKSON:def.9">Def9</a></i>;<br/>
<b>take </b><font color="Maroon" title="c13">i</font> = <font color="Maroon" title="c2">f</font> <a href="dickson.html#K7" title="DICKSON:func.7">mindex</a> <font color="Maroon" title="c12">b</font>;<br/>

<b>take </b>
<font color="Maroon" title="c9">mB</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
;<br/>

<a NAME="E20:47"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">BI</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E17:47"><i><font color="Green" title="E38">A8</font></i></a></i>;<br/>
<a NAME="E21:47"/><b>then </b>
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c8">BI</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E22:47"/>
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">mB</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>

<a NAME="E23:47"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17" title="NORMSP_1:th.17">NORMSP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E24:47"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">mB</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E17:47"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="dickson.html#E18:47"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="dickson.html#E19:47"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="dickson.html#D11" target="_self" title="DICKSON:def.11">Def11</a></i>;<br/>


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