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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#V2">admissible</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #);<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="bagorder.html#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #);<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E3:54"/>
 <a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E5:54"/><i><font color="Green">E45</font></i>: 
 <a href="bagorder.html#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #)</span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K13">FinOrd</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #)</span>)</span> #)
 
<b>by </b><i><a class="ref" href="bagorder.html#D18">BAGORDER:def 18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:54_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:54_1"/>
<font color="Maroon">c<sub>11</sub></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#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #)</span>)</span>
 <b>by </b><i><a class="txt" href="polyred.html#E5:54"><i><font color="Green">E45</font></i></a>, <a class="txt" href="polyred.html#E3:54_1"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:54"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #)</span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="bagorder.html#K15">MinElement</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><a href="bagorder.html#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #)</span>)</span>;<br/>
<a NAME="E8:54"/>
 <a href="bagorder.html#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #) is <a href="wellfnd1.html#V1">well_founded</a>
 
<b>by </b><i><a class="ref" href="bagorder.html#T42">BAGORDER:42</a></i>;<br/>
<a NAME="E9:54"/><b>then </b><i><font color="Green">E47</font></i>: 
(  <a href="bagorder.html#K15">MinElement</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><a href="bagorder.html#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #)</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="bagorder.html#K15">MinElement</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><a href="bagorder.html#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #)</span>)</span> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>10</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #)</span>)</span> )
 
<b>by </b><i><a class="txt" href="polyred.html#E7:54"><i><font color="Green">E46</font></i></a>, <a class="ref" href="bagorder.html#D19">BAGORDER:def 19</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E11:54"/><i><font color="Green">E48</font></i>: 
(  <a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15">MinElement</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><a href="bagorder.html#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #)</span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>assume </b><a NAME="E1:54_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E2:54_2"/><i><font color="Green">E50</font></i>: 
 <a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="polyred.html#E1:54_2"><i><font color="Green">E49</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_2_1_1"/>
(  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font> or  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1"><b>case </b></a><a NAME="E1:54_2_1_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="polyred.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_2"><b>case </b></a><a NAME="E1:54_2_1_1_2"/><i><font color="Green">E51</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:54_2_1_1_2"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K2" target="_self">Support</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K15">MinElement</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p3">(<span class="default"><a href="bagorder.html#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #)</span>)</span></span>)</span></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#K14">FinPoset</a> <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> #)</span>)</span>
 <b>by </b><i><a class="txt" href="polyred.html#E9:54"><i><font color="Green">E47</font></i></a>, <a class="txt" href="polyred.html#E11:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="polyred.html#E2:54_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="polyred.html#E1:54_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/><a NAME="E3:54_2_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polyred.html#E5:54"><i><font color="Green">E45</font></i></a>, <a class="txt" href="polyred.html#E11:54"><i><font color="Green">E48</font></i></a>, <a class="ref" href="polyred.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E4:54_2_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="polyred.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:54_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="polyred.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:54"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="polyred.html#R1" target="_self">&lt;=</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="polyred.html#E11:54"><i><font color="Green">E48</font></i></a></i>;<br/>


</div>
