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

<span class="kw">set </span><font color="Maroon" title="c5">BT</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">T</font> #);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">X</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#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">R</font> =  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>;<br/>
<a NAME="E1:79_1_1"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<a NAME="E2:79_1_1"/>
 <a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #) is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T42" title="BAGORDER:th.42">BAGORDER:42</a></span>;<br/>
<a NAME="E3:79_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A2</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#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #)</span>)</span> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellfnd1.html#D2" title="WELLFND1:def.2">WELLFND1:def 2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">Y</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="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> &amp; <font color="Maroon" title="c8">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  ex <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">p</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">Y</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:79_1_1_1"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
<font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:79_1_1_1"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
<font color="Maroon" title="c8">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">p</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">Y</font> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c9">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">Y</font>;<br/><a NAME="E4:79_1_1_1"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_1_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a></span>;<br/><a NAME="E5:79_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1_1_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a></span>;<br/><a NAME="E6:79_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A5</font></span>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D6" title="RELAT_1:def.6">RELAT_1:def 6</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:79_1_1_1_1_1"/>
( <font color="Maroon" title="c9">z</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> or <font color="Maroon" title="c9">z</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79_1_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:79_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">z</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</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="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:79_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:79_1_1_1_1_1_2"/>
<font color="Maroon" title="c9">z</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</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="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:79_1_1_1_1_1_2"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">z9</font> = <font color="Maroon" title="c9">z</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">M</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Olive" title="b1">y9</font></span>)</span> where <font color="Olive" title="b1">y9</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> :  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> &amp; <font color="Olive" title="b1">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> ) </span>}</span> ;<br/><a NAME="E9:79_1_1_1"/>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">z9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Olive" title="b1">y9</font></span>)</span> where <font color="Olive" title="b1">y9</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> :  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> &amp; <font color="Olive" title="b1">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_1_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">M</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Olive" title="b1">y9</font></span>)</span> where <font color="Olive" title="b1">y9</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> :  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> &amp; <font color="Olive" title="b1">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> ) </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">u</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="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font> implies <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:79_1_1_1_2"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</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="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #)</span>)</span></span><br/><a NAME="E2:79_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A6</font></span>: 
 ex <font color="Olive" title="b1">p</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <br/>(  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">u</font> &amp;  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> ) )
 ;<br/><a NAME="E3:79_1_1_1_2"/>
 <a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</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">T</font> #)</span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <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">T</font> #)</span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#D18" title="BAGORDER:def.18">BAGORDER:def 18</a></span>;<br/><span class="kw">hence </span><a NAME="E4:79_1_1_1_2"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_1_1_1_2"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</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><a NAME="E12:79_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:79_1_1_1"/><span class="lab"><font color="Green" title="E65">A7</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font>
 <span class="kw">and </span><br/><a NAME="E15:79_1_1_1"/><span class="lab"><font color="Green" title="E66">A8</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#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #)</span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c12">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_1_1"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">p</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E17:79_1_1_1"/><span class="lab"><font color="Green" title="E67">A9</font></span>: 
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font>
 <span class="kw">and </span><br/><a NAME="E18:79_1_1_1"/><span class="lab"><font color="Green" title="E68">A10</font></span>: 
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> &amp; <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:79_1_1_1"><span class="lab"><font color="Green" title="E65">A7</font></span></a></span>;<br/><a NAME="E19:79_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c14">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c15">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c14">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">Y</font>;<br/>
<a NAME="E2:79_1_1_1_3"/><span class="lab"><font color="Green" title="E69">A11</font></span>: 
 <a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</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">T</font> #)</span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K13" title="BAGORDER:func.13">FinOrd</a> <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">T</font> #)</span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#D18" title="BAGORDER:def.18">BAGORDER:def 18</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:79_1_1_1_3"/><span class="lab"><font color="Green" title="E70">A12</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c14">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:79_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c14">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:79_1_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c14">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></span>;<br/>
<a NAME="E6:79_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c15">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>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">h9</font>, <font color="Maroon" title="c17">g9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:79_1_1_1_3"/><span class="lab"><font color="Green" title="E72">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">h9</font>,<font color="Maroon" title="c17">g9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:79_1_1_1_3"/><span class="lab"><font color="Green" title="E73">A15</font></span>: 
<font color="Maroon" title="c16">h9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:79_1_1_1_3"/><span class="lab"><font color="Green" title="E74">A16</font></span>: 
<font color="Maroon" title="c17">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></span>;<br/>
<font color="Maroon" title="c15">b</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">h9</font>,<font color="Maroon" title="c17">g9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:79_1_1_1_3"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c17">g9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">b9</font> = <font color="Maroon" title="c15">b</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:79_1_1_1_3"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<span class="lab"><font color="Green" title="E75">A17</font></span>: <font color="Maroon" title="c14">p</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">h9</font>,<font color="Maroon" title="c17">g9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:79_1_1_1_3"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c16">h9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
;<br/>
<a NAME="E14:79_1_1_1_3"/>
not <font color="Maroon" title="c16">h9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1_1"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E9:79_1_1_1_3"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E15:79_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">h9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">p</font> = <font color="Maroon" title="c14">p</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:79_1_1_1_3"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/>
<a NAME="E17:79_1_1_1_3"/>
<font color="Maroon" title="c19">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c18">b9</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79_1_1_1_3"><span class="lab"><font color="Green" title="E71">A13</font></span></a>, <a class="ref" href="polyred.html#D13" target="_self" title="POLYRED:def.13">Def13</a></span>;<br/>
<a NAME="E18:79_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A18</font></span>: 
 ex <font color="Olive" title="b1">u</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c19">p</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c18">b9</font>,<font color="Olive" title="b1">u</font>,<font color="Maroon" title="c2">T</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D7" target="_self" title="POLYRED:def.7">Def7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">p</font> = <font color="Maroon" title="c19">p</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ;<br/>
<a NAME="E20:79_1_1_1_3"/><span class="lab"><font color="Green" title="E77">A19</font></span>: 
<font color="Maroon" title="c18">b9</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:79_1_1_1_3"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="ref" href="polyred.html#T43" target="_self" title="POLYRED:th.43">Th43</a></span>;<br/>
<a NAME="E21:79_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A20</font></span>: 
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c18">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c20">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D3" target="_self" title="POLYRED:def.3">Def3</a></span>;<br/>
<a NAME="E22:79_1_1_1_3"/>
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_1_1_1_3"><span class="lab"><font color="Green" title="E70">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E23:79_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A21</font></span>: 
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c18">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font>
 
;<br/>
<a NAME="E24:79_1_1_1_3"/>
<font color="Maroon" title="c18">b9</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:79_1_1_1_3"><span class="lab"><font color="Green" title="E77">A19</font></span></a>, <a class="ref" href="polyred.html#D3" target="_self" title="POLYRED:def.3">Def3</a></span>;<br/>
<a NAME="E25:79_1_1_1_3"/><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"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c18">b9</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c20">p</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> 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#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_1_1_1_3"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="ref" href="polyred.html#D2" target="_self" title="POLYRED:def.2">Def2</a></span>;<br/>
<a NAME="E26:79_1_1_1_3"/><span class="kw">then </span>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c18">b9</font> <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#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #)</span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c20">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:79_1_1_1_3"><span class="lab"><font color="Green" title="E78">A20</font></span></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></span>;<br/>
<a NAME="E27:79_1_1_1_3"/><span class="kw">then </span>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c18">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><a href="bagorder.html#K14" title="BAGORDER:func.14">FinPoset</a> <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">T</font> #)</span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:79_1_1_1"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="txt" href="#E23:79_1_1_1_3"><span class="lab"><font color="Green" title="E79">A21</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:79_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:79_1_1_1"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</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><a NAME="E20:79_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c14">p</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E21:79_1_1_1"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">p</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:79_1_1_1"><span class="lab"><font color="Green" title="E68">A10</font></span></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>
<a NAME="E5:79_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a>  is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#D2" title="WELLORD1:def.2">WELLORD1:def 2</a></span>;<br/>
<a NAME="E6:79_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A22</font></span>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> is  <a href="rewrite1.html#V1" title="REWRITE1:attr.1">co-well_founded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D13" title="REWRITE1:def.13">REWRITE1:def 13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">B</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>;<br/><span class="kw">let </span><font color="Maroon" title="c10">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> implies  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c10">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> )</span><br/><span class="kw">assume </span><a NAME="E1:79_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c10">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span><br/><a NAME="E2:79_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A23</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D6" title="RELAT_1:def.6">RELAT_1:def 6</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:79_1_1_2_1_1"/>
( <font color="Maroon" title="c10">x</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> <span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> or <font color="Maroon" title="c10">x</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> <span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_1_1_2"><span class="lab"><font color="Green" title="E63">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:79_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">x</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> <span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</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="c10">x</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span><br/><div class="add"><a NAME="E2:79_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:79_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">x</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:79_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">x</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> <span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</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="c10">x</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:79_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">x</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">x9</font> = <font color="Maroon" title="c10">x</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:79_1_1_2_2"/><span class="lab"><font color="Green" title="E64">A24</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c10">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:79_1_1_2_2"/><span class="lab"><font color="Green" title="E65">A25</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c10">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c13">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:79_1_1_2_2"/><span class="lab"><font color="Green" title="E66">A26</font></span>: 
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:79_1_1_2_2"/>
<font color="Maroon" title="c13">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1_1"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></span>;<br/><font color="Maroon" title="c10">x</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c13">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_1_1_2_2"><span class="lab"><font color="Green" title="E65">A25</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c12">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
;<br/><a NAME="E7:79_1_1_2_2"/><span class="kw">then </span>
not <font color="Maroon" title="c11">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_1_1_2_2"><span class="lab"><font color="Green" title="E66">A26</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E8:79_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">x9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">x9</font> = <font color="Maroon" title="c11">x9</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><a NAME="E10:79_1_1_2_2"/>
<font color="Maroon" title="c14">x9</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c14">x9</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1_1_2_2"><span class="lab"><font color="Green" title="E64">A24</font></span></a>, <a class="ref" href="polyred.html#D13" target="_self" title="POLYRED:def.13">Def13</a></span>;<br/><a NAME="E11:79_1_1_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c14">x9</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c14">x9</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D7" target="_self" title="POLYRED:def.7">Def7</a></span>;<br/><a NAME="E12:79_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">x9</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c14">x9</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T43" target="_self" title="POLYRED:th.43">Th43</a></span>;<br/><a NAME="E13:79_1_1_2_2"/><span class="kw">then </span>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c14">x9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c14">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D3" target="_self" title="POLYRED:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E14:79_1_1_2_2"/>
contradiction
 ; <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="E6:79_1_1_2"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c10">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 ; <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="E8:79_1_1"/><span class="kw">then </span>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> <a href="relat_2.html#R2" title="RELAT_2:pred.2">is_irreflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D2" title="RELAT_2:def.2">RELAT_2:def 2</a></span>;<br/>
<a NAME="E9:79_1_1"/><span class="kw">then </span>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> is  <a href="relat_2.html#V2" title="RELAT_2:attr.2">irreflexive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D10" title="RELAT_2:def.10">RELAT_2:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:79_1_1"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> is  <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79_1_1"><span class="lab"><font color="Green" title="E62">A22</font></span></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/>


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