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

<b>set </b><font color="Maroon">P</font> =  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">J</font> = <font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> ;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">SJ</font> =  <a href="pralg_1.html#K12">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span>;<br/>
<a NAME="E1:48_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/>for <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> holds <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> holds <br/><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </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="E1:48_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">g'</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E2:48_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font>
 ;<br/>

<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:48_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) )
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>thus </b><a NAME="E5:48_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>

<b>thus </b><a NAME="E6:48_1_1_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g'</font> holds <br/><font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g'</font>
 ;<br/>

<a NAME="E2:48_1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">R</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E4:48_1_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E5:48_1_1_1_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E6:48_1_1_1_1"/>
<font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, , <a class="txt" href="groeb_1.html#E1:48_1_1"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="dickson.html#D15">DICKSON:def 15</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> &amp;  ex <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E2:48_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E3:48_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2"><b>proof </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="E1:48_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:48_1_1_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E4:48_1_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  ) )
 
<b>by </b><i>, , </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E5:48_1_1_2"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E6:48_1_1_2"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:48_1_1_2"/><i><font color="Green">E77</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">n</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_2_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:48_1_1_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:48_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:48_1_1"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E6:48_1_1_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<a NAME="E11:48_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E13:48_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/>

<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E14:48_1_1_2"/>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b</font>
 ;<br/>

<b>thus </b><a NAME="E15:48_1_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:48_1_1"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">i</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:48_1_1"/><i><font color="Green">E80</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>set </b><font color="Maroon">R</font> =  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #);<br/>
<div><i><font color="Green">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">i</font>
 ;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_3"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">i</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:48_1_1_3"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> &amp;  ex <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> ) )
 <b>by </b><i>, <a class="txt" href="groeb_1.html#E1:48_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><b>hence </b><a NAME="E5:48_1_1_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:48_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:48_1_1"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #)  ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">x</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<i><font color="Green">E95</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> = 
<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><i><font color="Green">E97</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_4_2"><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="E1:48_1_1_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:48_1_1_4_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">L</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E4:48_1_1_4_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><a NAME="E5:48_1_1_4_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, , <a class="ref" href="dickson.html#D15">DICKSON:def 15</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E6:48_1_1_4_2"/>
<font color="Maroon">g</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:48_1_1_4"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E6:48_1_1_4"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>

<a NAME="E7:48_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font>]
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">g'</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:48_1_1_4"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> &amp;  ex <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) )
 ;<br/>
<b>thus </b><a NAME="E10:48_1_1_4"/>
<font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #);<br/><b>set </b><font color="Maroon">N'</font> = <font color="Maroon">i</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">N</font>;<br/><a NAME="E1:48_1_1_5"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">i</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/><b>then reconsider </b><font color="Maroon">N'</font> = <font color="Maroon">i</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">N</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</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="E4:48_1_1_5"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">B'</font> <a href="dickson.html#R1">is_Dickson-basis_of</a> <font color="Maroon">N'</font>, <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span> &amp; <font color="Maroon">B'</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="dickson.html#D10">DICKSON:def 10</a></i>;<br/><a NAME="E5:48_1_1_5"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N'</font> &amp; ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font> holds <br/> ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">a</font> ) ) )
 <b>by </b><i>, <a class="ref" href="dickson.html#D9">DICKSON:def 9</a></i>;<br/><b>set </b><font color="Maroon">B</font> = <font color="Maroon">i</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B'</font>;<br/><a NAME="E6:48_1_1_5"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">i</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B'</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T17">FINSET_1:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_5_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B'</font>
 ;<br/><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_5_1"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">i</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>thus </b><a NAME="E4:48_1_1_5_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div><a NAME="E8:48_1_1_5"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">i</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E9:48_1_1_5"/><i><font color="Green">E163</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span><br/> for <font color="Olive">ta</font>, <font color="Olive">tb</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">ta</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">tb</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> &amp; <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> holds <br/><font color="Olive">ta</font> <a href="polynom1.html#R3">divides</a> <font color="Olive">tb</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">ta</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>, <font color="Maroon">tb</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>;<br/>



<b>assume </b><a NAME="E1:48_1_1_5_2"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ta</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tb</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E2:48_1_1_5_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:48_1_1_5_2"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">ai</font>, <font color="Olive">bi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">ai</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">bi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">ai</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">bi</font> ) ) )
 <b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_5_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_5_2_1"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><b>then consider </b><font color="Maroon">R</font> being    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">ak</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">bk</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:48_1_1_5_2_1"/><i><font color="Green">E167</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">ak</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">bk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">ak</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">bk</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:48_1_1"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E4:48_1_1_5_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="dickson.html#K11">OrderedNAT</a> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:48_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E5:48_1_1_5_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ak</font>,<font color="Maroon">bk</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K10">NATOrd</a> 
 <b>by </b><i>, <a class="ref" href="dickson.html#D15">DICKSON:def 15</a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon">a'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:48_1_1_5_2_1"/><i><font color="Green">E168</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ak</font>,<font color="Maroon">bk</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b'</font> )
 <b>by </b><i><a class="ref" href="dickson.html#D14">DICKSON:def 14</a></i>;<br/><i><font color="Green">E169</font></i>: <font color="Maroon">a'</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ak</font>,<font color="Maroon">bk</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">ak</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/><font color="Maroon">b'</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ak</font>,<font color="Maroon">bk</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">bk</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><b>hence </b><a NAME="E10:48_1_1_5_2_1"/>
<font color="Maroon">ta</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">tb</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="groeb_1.html#E2:48_1_1"><i><font color="Green">E48</font></i></a>, , <a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:48_1_1_5_2"/>
<font color="Maroon">ta</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">tb</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T50">POLYNOM1:50</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:48_1_1_5"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #)  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/> ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #) st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B'</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_5_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #);<br/>

<b>assume </b><a NAME="E1:48_1_1_5_3"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<b>consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:48_1_1_5_3"/><i><font color="Green">E171</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:48_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E4:48_1_1_5_3"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> &amp;  ex <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> ) )
 
<b>by </b><i>, <a class="txt" href="groeb_1.html#E2:48_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E5:48_1_1_5_3"/>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, <a class="txt" href="groeb_1.html#E2:48_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">b'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:48_1_1_5_3"/><i><font color="Green">E173</font></i>: 
( <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> &amp; <font color="Maroon">b'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a'</font> )
 <b>by </b><i>, <a class="ref" href="dickson.html#D9">DICKSON:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font>;<br/>
<a NAME="E8:48_1_1_5_3"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">b1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E10:48_1_1_5_3"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E11:48_1_1_5_3"/>
<font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , <a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #) ;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E14:48_1_1_5_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B'</font>
 <b>by </b><i>, , <a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>

<b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a</font>, <font color="Maroon">bb</font> = <font color="Maroon">b</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> ;<br/>
<a NAME="E16:48_1_1_5_3"/>
<font color="Maroon">bb</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">aa</font>
 
<b>by </b><i>, <a class="txt" href="groeb_1.html#E2:48_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1"><i><font color="Green">E49</font></i></a>, , <a class="txt" href="groeb_1.html#E3"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/>
<a NAME="E17:48_1_1_5_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E18:48_1_1_5_3"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:48_1_1_5"/><b>then </b>
<font color="Maroon">i</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B'</font> <a href="dickson.html#R1">is_Dickson-basis_of</a> <font color="Maroon">N</font>, <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #)
 <b>by </b><i>, <a class="ref" href="dickson.html#D9">DICKSON:def 9</a></i>;<br/><b>hence </b><a NAME="E12:48_1_1_5"/>
 ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="dickson.html#R1">is_Dickson-basis_of</a> <font color="Maroon">N</font>, <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #) &amp; <font color="Olive">B</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:48_1_1"/>
 <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" target="_self">DivOrder</a> <font color="Maroon">n</font></span>)</span> #) is <a href="dickson.html#V4">Dickson</a>
 <b>by </b><i><a class="ref" href="dickson.html#D10">DICKSON:def 10</a></i>;<br/>


</div>
