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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">CNOB</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" target="_self">NaivelyOrderedBags</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">IROB</font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" target="_self">NaivelyOrderedBags</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">CP</font> = the <a href="struct_0.html#U1">carrier</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>set </b><font color="Maroon">IP</font> = the <a href="orders_2.html#U1">InternalRel</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/>
<a NAME="E1:56"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" target="_self">NaivelyOrderedBags</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:56"/><b>then </b><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" target="_self">NaivelyOrderedBags</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</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/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:56_1_1"/><i><font color="Green">E31</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#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" target="_self">NaivelyOrderedBags</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:56_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" target="_self">NaivelyOrderedBags</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" target="_self">NaivelyOrderedBags</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E3:56_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" target="_self">NaivelyOrderedBags</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" target="_self">NaivelyOrderedBags</a> <font color="Maroon">n</font></span>)</span> )
 ;<br/><a NAME="E4:56_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">b</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="bagorder.html#E1:56_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</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/>;<br/><a NAME="E6:56_1_1"/>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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/><a NAME="E7:56_1_1"/><b>then </b><i><font color="Green">E38</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>
 <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="56_1_1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">f</font> = <font color="Maroon">a'</font>;<br/><b>set </b><font color="Maroon">g</font> = <font color="Maroon">b'</font>;<br/><a NAME="E1:56_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">a'</font> is   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> &amp; <font color="Maroon">b</font> is   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E4:56_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">a'</font>, <font color="Maroon">g</font> = <font color="Maroon">b'</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="bagorder.html#E4:56_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>take </b><font color="Maroon">f</font> = <font color="Maroon">f</font>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">g</font>;<br/><b>thus </b><a NAME="E3:56_1_1_1"/>
( <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> )
 ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E4:56_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><b>set </b><font color="Maroon">R</font> = <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">i</font>;<br/><a NAME="E5:56_1_1_1"/><i><font color="Green">E50</font></i>: 
<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">i</font> <a href="hidden.html#R1">=</a>  <a href="dickson.html#K11">OrderedNAT</a> 
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon">R</font> = <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">i</font> as    <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>take </b><font color="Maroon">R</font> = <font color="Maroon">R</font>;<br/><b>set </b><font color="Maroon">xi</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/><b>set </b><font color="Maroon">yi</font> = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/><a NAME="E7:56_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E7:56_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E8:56_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E9:56_1_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E7:56_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/><a NAME="E10:56_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="txt" href="bagorder.html#E7:56_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E11:56_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E12:56_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E7:56_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/><b>then reconsider </b><font color="Maroon">xi</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>, <font color="Maroon">yi</font> = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i>, <a class="ref" href="bagorder.html#T3" target="_self">Th3</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/><b>take </b><font color="Maroon">xi</font> = <font color="Maroon">xi</font>;<br/><b>take </b><font color="Maroon">yi</font> = <font color="Maroon">yi</font>;<br/><b>thus </b><a NAME="E14:56_1_1_1"/>
( <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">i</font> &amp; <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/><b>reconsider </b><font color="Maroon">a''</font> = <font color="Maroon">a'</font>, <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="txt" href="bagorder.html#E4:56_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E16:56_1_1_1"/>
<font color="Maroon">a''</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b''</font>
 <b>by </b><i>, <a class="txt" href="bagorder.html#E1:56_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E17:56_1_1_1"/><b>then </b>
<font color="Maroon">a''</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b''</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D13">POLYNOM1:def 13</a></i>;<br/><a NAME="E18:56_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xi</font>,<font color="Maroon">yi</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#D14">DICKSON:def 14</a></i>;<br/><b>hence </b><a NAME="E19:56_1_1_1"/>
<font color="Maroon">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">yi</font>
 <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/></div><b>end;</b></div><a NAME="E9:56_1_1"/><b>then </b>
<font color="Maroon">a'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b'</font>
 <b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><b>hence </b><a NAME="E10:56_1_1"/>
<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#R2">in</a> the <a href="orders_2.html#U1">InternalRel</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="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:56_1"/><i><font color="Green">E68</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#R2">in</a> the <a href="orders_2.html#U1">InternalRel</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/><a NAME="E3:56_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="orders_2.html#U1">InternalRel</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> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="orders_2.html#U1">InternalRel</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="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E4:56_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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/><a NAME="E5:56_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <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="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a'</font>, <font color="Maroon">b2</font> = <font color="Maroon">b'</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</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/>;<br/><a NAME="E8:56_1"/>
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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/><a NAME="E9:56_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">a2</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="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><a NAME="E10:56_1"/>
<font color="Maroon">a2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b2</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<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="E12:56_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> )
 <b>and </b><br/><a NAME="E13:56_1"/><i><font color="Green">E84</font></i>: 
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">xi</font>, <font color="Olive">yi</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">xi</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">yi</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">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</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="56_1_2"><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="E1:56_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><b>consider </b><font color="Maroon">R</font> being    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">xi</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">yi</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:56_1_2"/><i><font color="Green">E86</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>
 <b>and </b><br/><a NAME="E4:56_1_2"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">xi</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">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>and </b><br/><a NAME="E5:56_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">yi</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:56_1_2"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="dickson.html#K11">OrderedNAT</a> 
 <b>by </b><i>, <a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E7:56_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xi</font>,<font color="Maroon">yi</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">xii</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">yii</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:56_1_2"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">xii</font>,<font color="Maroon">yii</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xi</font>,<font color="Maroon">yi</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E10:56_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">xii</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">yii</font>
 <b>by </b><i><a class="ref" href="dickson.html#D14">DICKSON:def 14</a></i>;<br/><a NAME="E11:56_1_2"/>
( <font color="Maroon">xii</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi</font> &amp; <font color="Maroon">yii</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yi</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E12:56_1_2"/>
<font color="Maroon">a'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><a NAME="E15:56_1"/><b>then </b>
<font color="Maroon">a'</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b'</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T50">POLYNOM1:50</a></i>;<br/><b>hence </b><a NAME="E16:56_1"/>
<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#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" target="_self">NaivelyOrderedBags</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:56"/>
 <a href="bagorder.html#K9" target="_self">NaivelyOrderedBags</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <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>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/>


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