<?xml version="1.0"?>
<div class="add">

<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="E1:47"/>
<font color="Maroon">R</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">R</font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">R</font>;<br/>

<b>set </b><font color="Maroon">N</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>;<br/>
<a NAME="E3:47"/><i><font color="Green">E29</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:47"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">B</font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>,<font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E6:47"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E8:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="dickson.html#E2:47"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="txt" href="dickson.html#E17:47"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">f</font> <a href="dickson.html#K7" target="_self">mindex</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">BI</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>] </span>}</span> ;<br/>
<a NAME="E8:47"/><i><font color="Green">E32</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</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>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/>
<a NAME="E10:47"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="dickson.html#K7" target="_self">mindex</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>] </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="dickson.html#K7" target="_self">mindex</a> <font color="Maroon">b</font>
 ;<br/><b>thus </b><a NAME="E4:47_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">BI</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>] </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="dickson.html#T2" target="_self">Th2</a>, , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">mB</font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">BI</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">j</font> = <font color="Maroon">mB</font> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon">fj</font> = <font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">mB</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<a NAME="E15:47"/>
<font color="Maroon">fj</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E10:47"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E17:47"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E18:47"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">fj</font>
 <b>by </b><i>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E19:47"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="dickson.html#T1" target="_self">Th1</a></i>;<br/>
<b>take </b><font color="Maroon">i</font> = <font color="Maroon">f</font> <a href="dickson.html#K7" target="_self">mindex</a> <font color="Maroon">b</font>;<br/>

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

<a NAME="E20:47"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BI</font>
 
<b>by </b><i/>;<br/>
<a NAME="E21:47"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">BI</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E22:47"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">mB</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>

<a NAME="E23:47"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17">NORMSP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E24:47"/>
<font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">mB</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="txt" href="dickson.html#E3:47"><i><font color="Green">E29</font></i></a></i>;<br/>


</div>
