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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font> being  <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">o</font> is  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a>  holds <br/>( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" title="BAGORDER:func.9">NaivelyOrderedBags</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">o</font> &amp; <font color="Olive" title="b1">o</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">o</font> be   <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">o</font> is  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a>  implies ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" title="BAGORDER:func.9">NaivelyOrderedBags</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c2">o</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:56"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c2">o</font> is  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" title="BAGORDER:func.9">NaivelyOrderedBags</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">o</font> &amp; <font color="Maroon" title="c2">o</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">n0</font> = <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">IRN</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" title="BAGORDER:func.9">NaivelyOrderedBags</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" title="BAGORDER:func.9">NaivelyOrderedBags</a> <font color="Maroon" title="c1">n</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font> )</span><br/><span class="kw">assume </span><a NAME="E1:56_1"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" title="BAGORDER:func.9">NaivelyOrderedBags</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font></span><br/><a NAME="E2:56_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" title="BAGORDER:func.9">NaivelyOrderedBags</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="relat_1.html#T20" title="RELAT_1:th.20">RELAT_1:20</a></span>;<br/><a NAME="E3:56_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" title="BAGORDER:func.9">NaivelyOrderedBags</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="relat_1.html#T20" title="RELAT_1:th.20">RELAT_1:20</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c6">b</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="bagorder.html#D13" target="_self" title="BAGORDER:def.13">Def13</a></span>;<br/><a NAME="E5:56_1"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c7">a1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c8">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="bagorder.html#D13" target="_self" title="BAGORDER:def.13">Def13</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">l</font> = <font color="Maroon" title="c8">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c7">a1</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:56_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span><br/><span class="lab"><font color="Green" title="E38">A5</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D6" title="PRE_POLY:def.6">PRE_POLY:def 6</a></span>
;<br/><a NAME="E3:56_1_1"/>
<font color="Maroon" title="c7">a1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="pre_poly.html#D11" title="PRE_POLY:def.11">PRE_POLY:def 11</a></span>;<br/><a NAME="E4:56_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c8">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font>

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:56_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c7">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>;<br/><a NAME="E8:56_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c7">a1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></span>;<br/><a NAME="E9:56_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c7">a1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c7">a1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E10:56_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="pre_poly.html#T53" title="PRE_POLY:th.53">PRE_POLY:53</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:56"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K9" title="BAGORDER:func.9">NaivelyOrderedBags</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">o</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">R</font> =  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n0</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">S</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">o</font> #);<br/>
<a NAME="E5:56"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">o</font> #) is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dickson.html#D3" title="DICKSON:def.3">DICKSON:def 3</a></span>;<br/>
<a NAME="E6:56"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</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="c3">n0</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">o</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="bagorder.html#T34" target="_self" title="BAGORDER:th.34">Th34</a></span>;<br/>
<a NAME="E7:56"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c3">n0</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> </span>)</span></span>)</span> <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 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">o</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T33" target="_self" title="BAGORDER:th.33">Th33</a></span>;<br/>
<a NAME="E8:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">o</font> #) <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E6:56"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="dickson.html#T50" title="DICKSON:th.50">DICKSON:50</a></span>;<br/>
<a NAME="E9:56"/>
<font color="Maroon" title="c2">o</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></span>;<br/>
<a NAME="E10:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T92" title="ORDERS_1:th.92">ORDERS_1:92</a></span>;<br/>
<a NAME="E11:56"/><span class="lab"><font color="Green" title="E40">A12</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></span>;<br/>
<a NAME="E12:56"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">o</font> #) is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="dickson.html#T17" title="DICKSON:th.17">DICKSON:17</a></span>;<br/>
<a NAME="E13:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A13</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:56"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="wellfnd1.html#D2" title="WELLFND1:def.2">WELLFND1:def 2</a></span>;<br/>
<a NAME="E14:56"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
<font color="Maroon" title="c2">o</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="#E11:56"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></span>;<br/>
<a NAME="E15:56"/>
<font color="Maroon" title="c2">o</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="wellord1.html#T5" title="WELLORD1:th.5">WELLORD1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:56"/>
<font color="Maroon" title="c2">o</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:56"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="ref" href="wellord1.html#D4" title="WELLORD1:def.4">WELLORD1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
