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

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:79_1_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="rewrite1.html#V1">co-well_founded</a> <a href="relat_2.html#V2">irreflexive</a> <a href="relat_1.html#NM1">Relation</a> holds <font color="Olive">b<sub>1</sub></font> is <a href="rewrite1.html#V3">strongly-normalizing</a>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:79_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E2:79_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D6">RELAT_1:def 6</a></i>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="struct_0.html#U1">carrier</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="polyred.html#E2:79_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:79_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:79_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</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="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:79_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> 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>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:79_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:79_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> 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>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</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> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_1_1_1_2"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:79_1_1_1_2"/><i><font color="Green">E63</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>10</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="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <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="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><font color="Maroon">c<sub>6</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="polyred.html#E3:79_1_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/><a NAME="E5:79_1_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="polyred.html#E3:79_1_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:79_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <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="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E8:79_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="polyred.html#R5" target="_self">reduces_to</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polyred.html#E1:79_1_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="polyred.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E9:79_1_1_1_2"/><b>then </b>
ex <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 <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="polyred.html#R4" target="_self">reduces_to</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E10:79_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="polyred.html#R2" 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="ref" href="polyred.html#T43" target="_self">Th43</a></i>;<br/><a NAME="E11:79_1_1_1_2"/><b>then </b>
 <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>12</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E12:79_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:79_1_1_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E3:79_1_1"/><b>then </b>
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R2">is_irreflexive_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D2">RELAT_2:def 2</a></i>;<br/>
<a NAME="E4:79_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="relat_2.html#V2">irreflexive</a>
 
<b>by </b><i><a class="ref" href="relat_2.html#D10">RELAT_2:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</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>7</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>;<br/>
<a NAME="E5:79_1_1"/>
 <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="E6:79_1_1"/><b>then </b><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="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="wellord1.html#R1">is_well_founded_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="ref" href="wellfnd1.html#D2">WELLFND1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:79_1_1_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>1</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> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E3:79_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="polyred.html#E1:79_1_1_2"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E4:79_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="polyred.html#E1:79_1_1_2"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E5:79_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D6">RELAT_1:def 6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_1_2_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> )
 </span><b>by </b><i><a class="txt" href="polyred.html#E5:79_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:79_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:79_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>10</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="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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:79_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:79_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>10</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="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</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="E8:79_1_1_2"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>1</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> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="polyred.html#E1:79_1_1_2"><i><font color="Green">E63</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Olive">b<sub>1</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> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:79_1_1_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</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="E3:79_1_1_2_2"/><i><font color="Green">E65</font></i>: 
(  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/><a NAME="E4:79_1_1_2_2"/>
 <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/><b>hence </b><a NAME="E5:79_1_1_2_2"/>
<font color="Maroon">c<sub>13</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#E3:79_1_1_2_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E11:79_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</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>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:79_1_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; 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> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="polyred.html#E6:79_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</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="E15:79_1_1_2"/><i><font color="Green">E66</font></i>: 
(  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="polyred.html#E13:79_1_1_2"><i><font color="Green">E65</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:79_1_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="polyred.html#E15:79_1_1_2"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E18:79_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_1_1_2_3"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E3:79_1_1_2_3"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>16</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#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="polyred.html#E1:79_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:79_1_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a>  )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E5:79_1_1_2_3"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:79_1_1_2_3"/><i><font color="Green">E71</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>17</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="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>18</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="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/>
<i><font color="Green">E72</font></i>: <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="polyred.html#E7:79_1_1_2_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E9:79_1_1_2_3"/>
not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="polyred.html#E7:79_1_1_2_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:79_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>14</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <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="txt" href="polyred.html#E8:79_1_1_2_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/>
<font color="Maroon">c<sub>16</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="polyred.html#E7:79_1_1_2_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>16</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="txt" href="polyred.html#E7:79_1_1_2_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E14:79_1_1_2_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="polyred.html#R5" target="_self">reduces_to</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polyred.html#E5:79_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="polyred.html#D13" target="_self">Def13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</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="E16:79_1_1_2_3"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="polyred.html#R4" target="_self">reduces_to</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7" target="_self">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>19</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <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/>
<a NAME="E18:79_1_1_2_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="polyred.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polyred.html#E16:79_1_1_2_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="polyred.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E19:79_1_1_2_3"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="polyred.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E20:79_1_1_2_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="polyred.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polyred.html#E18:79_1_1_2_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="polyred.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E21:79_1_1_2_3"/>
 <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/>
<a NAME="E22:79_1_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>22</sub></font></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#E20:79_1_1_2_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="polyred.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E23:79_1_1_2_3"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>20</sub></font> <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> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polyred.html#E19:79_1_1_2_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E24:79_1_1_2_3"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="polyred.html#E3:79_1_1_2_3"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E25:79_1_1_2_3"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<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="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="polyred.html#E15:79_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="polyred.html#E23:79_1_1_2_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E26:79_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="polyred.html#E13:79_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:79_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E20:79_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="polyred.html#E17:79_1_1_2"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:79_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K6">~</a>  is <a href="wellord1.html#V1">well_founded</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#D2">WELLORD1:def 2</a></i>;<br/>
<a NAME="E9:79_1_1"/><b>then </b>
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V1">co-well_founded</a>
 
<b>by </b><i><a class="ref" href="rewrite1.html#D13">REWRITE1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:79_1_1"/>
 <a href="polyred.html#K3" target="_self">PolyRedRel</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V3">strongly-normalizing</a>
 <b>by </b><i><a class="txt" href="polyred.html#E1:79_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="polyred.html#E4:79_1_1"><i><font color="Green">E61</font></i></a></i>;<br/>


</div>
