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

<b>set </b><font color="Maroon" title="c2">P</font> =  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c3">J</font> = <font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> ;<br/>
<b>set </b><font color="Maroon" title="c4">S</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">SJ</font> =  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span>;<br/>
<a NAME="E1:48_1_1"/><i><font color="Green" title="E35">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/>for <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> holds <br/>(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b2">g</font> holds <br/><font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1_1"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">g'</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>

<b>assume </b><a NAME="E2:48_1_1_1"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g'</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:48_1_1_1"/><i><font color="Green" title="E38">A4</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:48_1_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>thus </b><a NAME="E5:48_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:48_1_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="groeb_1.html#E4:48_1_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>

<b>thus </b><a NAME="E6:48_1_1_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g'</font> holds <br/><font color="Maroon" title="c7">g'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1_1_1"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g'</font>
 ;<br/>

<a NAME="E2:48_1_1_1_1"/><b>then </b><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E2:48_1_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="groeb_1.html#E4:48_1_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">R</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E4:48_1_1_1_1"/><i><font color="Green" title="E41">A7</font></i>: 
( <font color="Maroon" title="c10">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</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="c10">R</font> )
 <b>by </b><i><a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E5:48_1_1_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">R</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E2:48_1_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="groeb_1.html#E4:48_1_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="groeb_1.html#E1:48_1_1_1_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="groeb_1.html#E4:48_1_1_1_1"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:48_1_1_1_1"/>
<font color="Maroon" title="c7">g'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</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="groeb_1.html#E2:48_1_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="groeb_1.html#E4:48_1_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="groeb_1.html#E2:48_1_1_1_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="groeb_1.html#E4:48_1_1_1_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="dickson.html#D15" title="DICKSON:def.15">DICKSON:def 15</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">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" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> ) );<br/>
<a NAME="E2:48_1_1"/><i><font color="Green" title="E36">A8</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
;<br/>
<a NAME="E3:48_1_1"/><i><font color="Green" title="E37">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1_2"/><i><font color="Green" title="E38">A10</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:48_1_1_2"/><i><font color="Green" title="E39">A11</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/><font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E4:48_1_1_2"/><i><font color="Green" title="E40">A12</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> holds <br/><font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</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="groeb_1.html#E1:48_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="groeb_1.html#E1:48_1_1_2"><i><font color="Green" title="E38">A10</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1_2"><i><font color="Green" title="E39">A11</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E5:48_1_1_2"/><i><font color="Green" title="E41">A13</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
;<br/>
<a NAME="E6:48_1_1_2"/><i><font color="Green" title="E42">A14</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c8">b</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:48_1_1_2"/><i><font color="Green" title="E43">A15</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="groeb_1.html#E5:48_1_1_2"><i><font color="Green" title="E41">A13</font></i></a>, <a class="txt" href="groeb_1.html#E6:48_1_1_2"><i><font color="Green" title="E42">A14</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c8">b</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="groeb_1.html#E8:48_1_1_2"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">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" title="c10">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">b</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_2_1"/><i><font color="Green" title="E44">A16</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:48_1_1_2_1"/><i><font color="Green" title="E45">A17</font></i>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E8:48_1_1_2"><i><font color="Green" title="E43">A15</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1_2_1"><i><font color="Green" title="E44">A16</font></i></a></i>;<br/><a NAME="E5:48_1_1_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:48_1_1_2"><i><font color="Green" title="E40">A12</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1_2_1"><i><font color="Green" title="E44">A16</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E6:48_1_1_2_1"/>
<font color="Maroon" title="c10">u</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="groeb_1.html#E1:48_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="groeb_1.html#E1:48_1_1_2"><i><font color="Green" title="E38">A10</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1_2"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="groeb_1.html#E4:48_1_1_2_1"><i><font color="Green" title="E45">A17</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:48_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<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="c10">b</font> = <font color="Maroon" title="c9">b</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="seqm_3.html#D8" title="SEQM_3:def.8">SEQM_3:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">b</font>
;<br/>

<b>thus </b><a NAME="E13:48_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:48_1_1_2"><i><font color="Green" title="E38">A10</font></i></a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c10">b</font>
;<br/>

<b>thus </b><a NAME="E14:48_1_1_2"/>
<font color="Maroon" title="c10">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c10">b</font>
 ;<br/>

<b>thus </b><a NAME="E15:48_1_1_2"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:48_1_1_2"><i><font color="Green" title="E40">A12</font></i></a>, <a class="txt" href="groeb_1.html#E8:48_1_1_2"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">i</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:48_1_1"/><i><font color="Green" title="E38">A18</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="groeb_1.html#E2:48_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1"><i><font color="Green" title="E37">A9</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c7">R</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/>
<div><i><font color="Green" title="E39">A19</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" title="c8">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_3"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">i</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_3"/><i><font color="Green" title="E40">A20</font></i>: 
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:48_1_1_3"/>
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">u</font> ) )
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:48_1_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1_3"><i><font color="Green" title="E40">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E5:48_1_1_3"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:48_1_1_3"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:48_1_1"/><i><font color="Green" title="E40">A21</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #)  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">x</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<i><font color="Green" title="E41">A22</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> = 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<div><i><font color="Green" title="E42">A23</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" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_4_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:48_1_1_4_2"/><b>then </b><i><font color="Green" title="E43">A24</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">L</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E4:48_1_1_4_2"/><i><font color="Green" title="E44">A25</font></i>: 
( <font color="Maroon" title="c11">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</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="c11">L</font> )
 <b>by </b><i><a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><a NAME="E5:48_1_1_4_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">L</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="txt" href="groeb_1.html#E2:48_1_1_4_2"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="groeb_1.html#E4:48_1_1_4_2"><i><font color="Green" title="E44">A25</font></i></a>, <a class="ref" href="dickson.html#D15" title="DICKSON:def.15">DICKSON:def 15</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E6:48_1_1_4_2"/>
<font color="Maroon" title="c9">g</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:48_1_1_4_2"><i><font color="Green" title="E44">A25</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:48_1_1_4"/><b>then </b><i><font color="Green" title="E43">A26</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E2:48_1_1_4"><i><font color="Green" title="E41">A22</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">g</font>
;<br/>

<b>thus </b><a NAME="E6:48_1_1_4"/>
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:48_1_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="groeb_1.html#E2:48_1_1_4"><i><font color="Green" title="E41">A22</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1_4"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">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" title="c10">g</font>,<font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">g</font>]
 
<b>by </b><i><a class="txt" href="groeb_1.html#E5:48_1_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="groeb_1.html#E4:48_1_1_4"><i><font color="Green" title="E43">A26</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">g'</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:48_1_1_4"/><i><font color="Green" title="E44">A27</font></i>: 
( <font color="Maroon" title="c11">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> &amp; <font color="Maroon" title="c11">g'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">g</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> ) )
 ;<br/>
<b>thus </b><a NAME="E10:48_1_1_4"/>
<font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:48_1_1_4"><i><font color="Green" title="E44">A27</font></i></a></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" title="c8">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/><b>set </b><font color="Maroon" title="c9">N'</font> = <font color="Maroon" title="c6">i</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c8">N</font>;<br/><a NAME="E1:48_1_1_5"/><i><font color="Green" title="E41">A28</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c8">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:48_1_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="relat_1.html#T167" title="RELAT_1:th.167">RELAT_1:167</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">N'</font> = <font color="Maroon" title="c6">i</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c8">N</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">B'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:48_1_1_5"/><i><font color="Green" title="E42">A29</font></i>: 
( <font color="Maroon" title="c11">B'</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c10">N'</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span> &amp; <font color="Maroon" title="c11">B'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="ref" href="dickson.html#D10" title="DICKSON:def.10">DICKSON:def 10</a></i>;<br/><a NAME="E5:48_1_1_5"/><i><font color="Green" title="E43">A30</font></i>: 
( <font color="Maroon" title="c11">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">N'</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N'</font> holds <br/> ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B'</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> ) ) )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:48_1_1_5"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="dickson.html#D9" title="DICKSON:def.9">DICKSON:def 9</a></i>;<br/><b>set </b><font color="Maroon" title="c12">B</font> = <font color="Maroon" title="c6">i</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">B'</font>;<br/><a NAME="E6:48_1_1_5"/><i><font color="Green" title="E44">A31</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">B'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:48_1_1_5"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="finset_1.html#T17" title="FINSET_1:th.17">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" title="c13">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_5_1"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">i</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">B'</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_5_1"/><i><font color="Green" title="E45">A32</font></i>: 
( <font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B'</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>thus </b><a NAME="E4:48_1_1_5_1"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:48_1_1_5"><i><font color="Green" title="E43">A30</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1_5_1"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">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" title="E45">A33</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">N</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E9:48_1_1_5"/><i><font color="Green" title="E46">A34</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span><br/> for <font color="Olive" title="b3">ta</font>, <font color="Olive" title="b4">tb</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">ta</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">tb</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b3">ta</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b4">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" title="c13">a</font>, <font color="Maroon" title="c14">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c15">ta</font>, <font color="Maroon" title="c16">tb</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>;<br/>



<b>assume </b><a NAME="E1:48_1_1_5_2"/><i><font color="Green" title="E47">A35</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">ta</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">tb</font> &amp; <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E2:48_1_1_5_2"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">f</font>, <font color="Maroon" title="c18">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:48_1_1_5_2"/><i><font color="Green" title="E48">A36</font></i>: 
( <font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> &amp; <font color="Maroon" title="c18">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">ai</font>, <font color="Olive" title="b4">bi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">bi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">ai</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">bi</font> ) ) )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:48_1_1_5_2"><i><font color="Green" title="E47">A35</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">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" title="c19">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_5_2_1"/><i><font color="Green" title="E49">A37</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c20">R</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> , <font color="Maroon" title="c21">ak</font>, <font color="Maroon" title="c22">bk</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c20">R</font><b> such that </b><br/><a NAME="E3:48_1_1_5_2_1"/><i><font color="Green" title="E50">A38</font></i>: 
( <font color="Maroon" title="c20">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> &amp; <font color="Maroon" title="c21">ak</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> &amp; <font color="Maroon" title="c22">bk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> &amp; <font color="Maroon" title="c21">ak</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c22">bk</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:48_1_1_5_2"><i><font color="Green" title="E48">A36</font></i></a></i>;<br/><a NAME="E4:48_1_1_5_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:48_1_1_5_2_1"><i><font color="Green" title="E49">A37</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">ak</font>,<font color="Maroon" title="c22">bk</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K10" title="DICKSON:func.10">NATOrd</a> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:48_1_1_5_2_1"><i><font color="Green" title="E50">A38</font></i></a>, <a class="ref" href="dickson.html#D15" title="DICKSON:def.15">DICKSON:def 15</a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">a'</font>, <font color="Maroon" title="c24">b'</font> being    <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> such that </b><br/><a NAME="E7:48_1_1_5_2_1"/><i><font color="Green" title="E51">A39</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a'</font>,<font color="Maroon" title="c24">b'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">ak</font>,<font color="Maroon" title="c22">bk</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c23">a'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">b'</font> )
 <b>by </b><i><a class="ref" href="dickson.html#D14" title="DICKSON:def.14">DICKSON:def 14</a></i>;<br/><i><font color="Green" title="E52">A40</font></i>: <font color="Maroon" title="c23">a'</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">ak</font>,<font color="Maroon" title="c22">bk</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E7:48_1_1_5_2_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c21">ak</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>
;<br/><font color="Maroon" title="c24">b'</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">ak</font>,<font color="Maroon" title="c22">bk</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E7:48_1_1_5_2_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c22">bk</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/><b>hence </b><a NAME="E10:48_1_1_5_2_1"/>
<font color="Maroon" title="c15">ta</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">tb</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:48_1_1_5_2"><i><font color="Green" title="E47">A35</font></i></a>, <a class="txt" href="groeb_1.html#E4:48_1_1_5_2"><i><font color="Green" title="E48">A36</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1_5_2_1"><i><font color="Green" title="E50">A38</font></i></a>, <a class="txt" href="groeb_1.html#E7:48_1_1_5_2_1"><i><font color="Green" title="E51">A39</font></i></a>, <a class="txt" href="groeb_1.html#E8:48_1_1_5_2_1"><i><font color="Green" title="E52">A40</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:48_1_1_5_2"/>
<font color="Maroon" title="c15">ta</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c16">tb</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T50" title="POLYNOM1:th.50">POLYNOM1:50</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:48_1_1_5"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #)  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> holds <br/> ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #) st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">i</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">B'</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">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" title="c13">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/>

<b>assume </b><a NAME="E1:48_1_1_5_3"/><i><font color="Green" title="E47">A41</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c14">a'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:48_1_1_5_3"/><i><font color="Green" title="E48">A42</font></i>: 
( <font color="Maroon" title="c14">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E7:48_1_1"><i><font color="Green" title="E40">A21</font></i></a></i>;<br/>
<a NAME="E4:48_1_1_5_3"/><i><font color="Green" title="E49">A43</font></i>: 
( <font color="Maroon" title="c14">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> &amp;  ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a'</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a'</font> ) )
 
<b>by </b><i><a class="txt" href="groeb_1.html#E5:48_1_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1_5_3"><i><font color="Green" title="E48">A42</font></i></a></i>;<br/>
<a NAME="E5:48_1_1_5_3"/>
<font color="Maroon" title="c14">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">N'</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E1:48_1_1_5_3"><i><font color="Green" title="E47">A41</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1_5_3"><i><font color="Green" title="E48">A42</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">b'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:48_1_1_5_3"/><i><font color="Green" title="E50">A44</font></i>: 
( <font color="Maroon" title="c15">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B'</font> &amp; <font color="Maroon" title="c15">b'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">a'</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:48_1_1_5"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="dickson.html#D9" title="DICKSON:def.9">DICKSON:def 9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b'</font>;<br/>
<a NAME="E8:48_1_1_5_3"/><i><font color="Green" title="E51">A45</font></i>: 
<font color="Maroon" title="c11">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E1:48_1_1_5"><i><font color="Green" title="E41">A28</font></i></a>, <a class="txt" href="groeb_1.html#E5:48_1_1_5"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">b1</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E10:48_1_1_5_3"/><i><font color="Green" title="E52">A46</font></i>: 
( <font color="Maroon" title="c17">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b'</font> &amp; <font color="Maroon" title="c17">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b'</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:48_1_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="groeb_1.html#E7:48_1_1_5_3"><i><font color="Green" title="E50">A44</font></i></a></i>;<br/>
<a NAME="E11:48_1_1_5_3"/>
<font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E5:48_1_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="groeb_1.html#E7:48_1_1_5_3"><i><font color="Green" title="E50">A44</font></i></a>, <a class="txt" href="groeb_1.html#E8:48_1_1_5_3"><i><font color="Green" title="E51">A45</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">b</font> = <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b'</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="groeb_1.html#E6:48_1_1"><i><font color="Green" title="E39">A19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">b</font> = <font color="Maroon" title="c18">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #) ;<br/>
<b>take </b>
<font color="Maroon" title="c19">b</font>
;<br/>

<b>thus </b><a NAME="E14:48_1_1_5_3"/>
<font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">i</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">B'</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:48_1_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="groeb_1.html#E7:48_1_1_5_3"><i><font color="Green" title="E50">A44</font></i></a>, <a class="txt" href="groeb_1.html#E8:48_1_1_5_3"><i><font color="Green" title="E51">A45</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c20">aa</font> = <font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c21">bb</font> = <font color="Maroon" title="c19">b</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/>
<a NAME="E16:48_1_1_5_3"/>
<font color="Maroon" title="c21">bb</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c20">aa</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E9:48_1_1_5"><i><font color="Green" title="E46">A34</font></i></a>, <a class="txt" href="groeb_1.html#E3:48_1_1_5_3"><i><font color="Green" title="E48">A42</font></i></a>, <a class="txt" href="groeb_1.html#E4:48_1_1_5_3"><i><font color="Green" title="E49">A43</font></i></a>, <a class="txt" href="groeb_1.html#E7:48_1_1_5_3"><i><font color="Green" title="E50">A44</font></i></a>, <a class="txt" href="groeb_1.html#E8:48_1_1_5_3"><i><font color="Green" title="E51">A45</font></i></a>, <a class="txt" href="groeb_1.html#E10:48_1_1_5_3"><i><font color="Green" title="E52">A46</font></i></a></i>;<br/>
<a NAME="E17:48_1_1_5_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">b</font>,<font color="Maroon" title="c13">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>  <a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="groeb_1.html#D5" target="_self" title="GROEB_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E18:48_1_1_5_3"/>
<font color="Maroon" title="c19">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">a</font>
 <b>by </b><i><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><a NAME="E11:48_1_1_5"/><b>then </b>
<font color="Maroon" title="c6">i</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c11">B'</font> <a href="dickson.html#R1" title="DICKSON:pred.1">is_Dickson-basis_of</a> <font color="Maroon" title="c8">N</font>, <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 <b>by </b><i><a class="txt" href="groeb_1.html#E8:48_1_1_5"><i><font color="Green" title="E45">A33</font></i></a>, <a class="ref" href="dickson.html#D9" title="DICKSON:def.9">DICKSON:def 9</a></i>;<br/><b>hence </b><a NAME="E12:48_1_1_5"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <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="c8">N</font>, <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #) &amp; <font color="Olive" title="b1">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E6:48_1_1_5"><i><font color="Green" title="E44">A31</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:48_1_1"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="groeb_1.html#K4" title="GROEB_1:func.4">DivOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 <b>by </b><i><a class="ref" href="dickson.html#D10" title="DICKSON:def.10">DICKSON:def 10</a></i>;<br/>


</div>
