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<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>let </b><font color="Maroon">o</font> be   <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/>



<b>assume </b><a NAME="E1:57"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">o</font> is <a href="bagorder.html#V2" target="_self">admissible</a>
 ;<br/>

<b>set </b><font color="Maroon">IRN</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_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/><b>assume </b><a NAME="E1:57_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="E2:57_1"/>
( <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/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> <b>by </b><i><a class="txt" href="bagorder.html#E1:57_1"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E4:57_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <a class="txt" href="bagorder.html#E1:57_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>set </b><font color="Maroon">l</font> = <font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">a1</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:57_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><i><font color="Green">E38</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">a1</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">a1</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">a1</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D6">POLYNOM1:def 6</a></i>
;<br/><a NAME="E3:57_1_1"/>
<font color="Maroon">a1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:57_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="polynom1.html#D13">POLYNOM1:def 13</a></i>;<br/><a NAME="E4:57_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">a1</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">a1</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>

;<br/><br/></div><b>end;</b></div><a NAME="E6:57_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">a1</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">a1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/><a NAME="E7:57_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">a1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o</font>
 <b>by </b><i>, </i>;<br/><a NAME="E8:57_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">a1</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">a1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:57_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> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T57">POLYNOM1:57</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:57"/><i><font color="Green">E49</font></i>: 
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> <a href="tarski.html#R1">c=</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>

<b>set </b><font color="Maroon">R</font> =  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">o</font> #);<br/>
<a NAME="E4:57"/><i><font color="Green">E50</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">o</font> #) is <a href="dickson.html#V3">quasi_ordered</a>
 
<b>by </b><i><a class="ref" href="dickson.html#D3">DICKSON:def 3</a></i>;<br/>
<a NAME="E5:57"/><i><font color="Green">E62</font></i>: 
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> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">o</font> #)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:57"/>
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> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">o</font> #)
 
<b>by </b><i/>;<br/>
<a NAME="E7:57"/><b>then </b><i><font color="Green">E64</font></i>: 
<a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">o</font> #) <a href="dickson.html#K5">\~</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 
<b>by </b><i>, , <a class="ref" href="dickson.html#T50">DICKSON:50</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><a NAME="E1:57_2"/>
<font color="Maroon">o</font> <a href="relat_2.html#R7">is_strongly_connected_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E2:57_2"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">o</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">o</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="orders_1.html#T92">ORDERS_1:92</a></i>;<br/><a NAME="E3:57_2"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/><b>thus </b><a NAME="E4:57_2"/>
<font color="Maroon">o</font> is <a href="relat_2.html#V1">reflexive</a>
 ;<br/><b>thus </b><a NAME="E5:57_2"/>
<font color="Maroon">o</font> is <a href="relat_2.html#V8">transitive</a>
 ;<br/><b>thus </b><a NAME="E6:57_2"/>
<font color="Maroon">o</font> is <a href="relat_2.html#V4">antisymmetric</a>
 ;<br/><b>thus </b><a NAME="E7:57_2"/>
<font color="Maroon">o</font> is <a href="relat_2.html#V6">connected</a>
 <b>by </b><i>, , <a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a></i>;<br/><a NAME="E8:57_2"/>
 <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">o</font> #) is <a href="wellfnd1.html#V1">well_founded</a>
 <b>by </b><i>, <a class="ref" href="dickson.html#T17">DICKSON:17</a></i>;<br/><a NAME="E9:57_2"/><b>then </b>
<font color="Maroon">o</font> <a href="wellord1.html#R1">is_well_founded_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="wellfnd1.html#D2">WELLFND1:def 2</a></i>;<br/><b>hence </b><a NAME="E10:57_2"/>
<font color="Maroon">o</font> is <a href="wellord1.html#V1">well_founded</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T5">WELLORD1:5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:57"/>
<font color="Maroon">o</font> is <a href="wellord1.html#V2">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#D4">WELLORD1:def 4</a></i>;<br/>


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