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

<b>let </b><font color="Maroon">o</font> be  <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>set </b><font color="Maroon">R</font> =  <a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font>;<br/>
<b>set </b><font color="Maroon">r</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">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #);<br/>
<b>set </b><font color="Maroon">ir</font> = the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #);<br/>
<b>set </b><font color="Maroon">cr</font> = 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">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #);<br/>
<b>assume </b><a NAME="E1:48"/>
 <a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font> is <a href="wellord1.html#V2">well-ordering</a>
 ;<br/>

<a NAME="E2:48"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font> is <a href="wellord1.html#V1">well_founded</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#D4">WELLORD1:def 4</a></i>;<br/>
<a NAME="E3:48"/>
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">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="orders_1.html#T100">ORDERS_1:100</a></i>;<br/>
<a NAME="E4:48"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #) <a href="wellord1.html#R1">is_well_founded_in</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">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #)
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T5">WELLORD1:5</a></i>;<br/>
<a NAME="E5:48"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #) is <a href="wellfnd1.html#V1">well_founded</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#D2">WELLFND1:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">a<sub>1</sub></font>,1;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">n</font>,1;<br/><a NAME="E1:48_1"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">n</font>,1 as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">o</font> ;<br/><a NAME="E3:48_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o</font> )
 <b>by </b><i><a class="ref" href="card_4.html#T8">CARD_4:8</a></i>;<br/><a NAME="E4:48_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_7.html#D3">FUNCT_7:def 3</a></i>;<br/><a NAME="E5:48_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/><a NAME="E6:48_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/><a NAME="E7:48_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="finsub_1.html#K1">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E8:48_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:48_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">y</font> is <a href="seqm_3.html#V6">natural-yielding</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:48_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_1"/>
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><b>that </b><br/><a NAME="E2:48_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E3:48_1_1"/><i><font color="Green">E62</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E4:48_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E6:48"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:48_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_2_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E11:48_1"/><b>then </b>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E12:48_1"/><b>then </b>
<font color="Maroon">y</font> is <a href="polynom1.html#V1">finite-support</a>
 <b>by </b><i><a class="ref" href="polynom1.html#D8">POLYNOM1:def 8</a></i>;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of 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">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #) <b>by </b><i>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E14:48_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">y</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</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">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #)<b> such that </b><br/><a NAME="E8:48"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="bagorder.html#E5:48"><i><font color="Green">E31</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="normsp_1.html#NM2">sequence</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">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #) ;<br/>
<a NAME="E10:48"/>
<font color="Maroon">f</font> is <a href="wellfnd1.html#V3">descending</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a><br/>

<b>set </b><font color="Maroon">fn1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">fn</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>;<br/>
<a NAME="E1:48_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1
 
<b>by </b><i/>;<br/>
<a NAME="E2:48_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="polynom1.html#K2">+*</a> <font color="Maroon">n</font>,1
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">fn1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fn</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E5:48_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T8">CARD_4:8</a></i>;<br/>
<a NAME="E6:48_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<b>then </b><i><font color="Green">E71</font></i>: <font color="Maroon">fn1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E8:48_2"/><i><font color="Green">E84</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E9:48_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">fn</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:48_2_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:48_2_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E3:48_2_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:48_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 ;<br/><a NAME="E5:48_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/><a NAME="E6:48_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><a NAME="E7:48_2_2"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E8:48_2_2"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>hence </b><font color="Maroon">fn1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font>
<b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">fn</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">l</font>
<b>by </b><i>, , <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E11:48_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">fn1</font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">fn</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E6:48"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="polynom1.html#D11">POLYNOM1:def 11</a></i>;<br/>
<b>thus </b><a NAME="E12:48_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="txt" href="bagorder.html#E6:48"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>

<a NAME="E13:48_2"/>
<font color="Maroon">fn1</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">fn</font>
 
<b>by </b><i>, <a class="ref" href="polynom1.html#D12">POLYNOM1:def 12</a></i>;<br/>
<a NAME="E14:48_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#NK4" target="_self">LexOrder</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D16">POLYNOM1:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon">tn</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:48_2"/><i><font color="Green">E111</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">fn</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">tn</font>
 <b>and </b><br/><a NAME="E17:48_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">tn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fn</font> <a href="polynom2.html#K2">*</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">fn</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">tn1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:48_2"/><i><font color="Green">E125</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">fn1</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">tn1</font>
 <b>and </b><br/><a NAME="E20:48_2"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">tn1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fn1</font> <a href="polynom2.html#K2">*</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">fn1</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E21:48_2"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o</font> )
 
<b>by </b><i><a class="ref" href="card_4.html#T8">CARD_4:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">nn</font> = <font color="Maroon">n</font>, <font color="Maroon">n1n</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">o</font> <b>by </b><i/>;<br/>
<a NAME="E23:48_2"/><i><font color="Green">E154</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E24:48_2"/>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">o</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#T7">WELLORD2:7</a></i>;<br/>
<a NAME="E25:48_2"/><b>then </b>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">o</font> is <a href="orders_1.html#V3">being_linear-order</a>
 
<b>by </b><i><a class="ref" href="orders_1.html#T107">ORDERS_1:107</a></i>;<br/>
<a NAME="E26:48_2"/><b>then </b><i><font color="Green">E155</font></i>: 
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">o</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">o</font>
 
<b>by </b><i>, <a class="ref" href="orders_1.html#T133">ORDERS_1:133</a></i>;<br/>
<a NAME="E27:48_2"/><b>then </b><i><font color="Green">E156</font></i>: 
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">o</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">fn</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/>
<a NAME="E28:48_2"/><i><font color="Green">E157</font></i>: 
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">o</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">fn1</font>
 
<b>by </b><i><a class="ref" href="bagorder.html#T4" target="_self">Th4</a>, <a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_2_3_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">fn1</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n1n</font>
 ;<br/><a NAME="E2:48_2_3_1_1"/><b>then </b>
<font color="Maroon">fn1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:48_2_3_1_1"/><b>then </b>
<font color="Maroon">fn1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E4:48_2_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:48_2_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n1n</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:48_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n1n</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E3:48_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:48_2_3"/><b>then </b>
<font color="Maroon">fn1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E5:48_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">fn1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E30:48_2"/><b>then </b>
 <a href="polynom2.html#K1">support</a> <font color="Maroon">fn1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n1n</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E31:48_2"/><b>then </b><i><font color="Green">E159</font></i>: 
 <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">fn1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n1n</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E32:48_2"/><i><font color="Green">E160</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fn1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_2_4_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">fn</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2_4_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">nn</font>
 ;<br/><a NAME="E2:48_2_4_1_1"/><b>then </b>
<font color="Maroon">fn</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:48_2_4_1_1"/><b>then </b>
<font color="Maroon">fn</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="bagorder.html#T5" target="_self">Th5</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E4:48_2_4_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="bagorder.html#T5" target="_self">Th5</a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:48_2_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">nn</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:48_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">nn</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E3:48_2_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">nn</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:48_2_4"/><b>then </b>
<font color="Maroon">fn</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E5:48_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">fn</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E34:48_2"/><b>then </b>
 <a href="polynom2.html#K1">support</a> <font color="Maroon">fn</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">nn</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E35:48_2"/><b>then </b>
 <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">fn</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">nn</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><i><font color="Green">E162</font></i>: <font color="Maroon">tn</font> = 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fn</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, , , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fn1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n1n</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
<br/>.= 
<font color="Maroon">tn1</font>
<b>by </b><i>, , , </i>
;<br/>
<a NAME="E37:48_2"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#NK4" target="_self">LexOrder</a> <font color="Maroon">o</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="polynom1.html#K9">+</a> <font color="Olive">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b</font> <a href="polynom1.html#K9">+</a> <font color="Olive">c</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#NK4" target="_self">LexOrder</a> <font color="Maroon">o</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E38:48_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K6" target="_self">GrLexOrder</a> <font color="Maroon">o</font></span>)</span> #)
 <b>by </b><i>, <a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, , <a class="ref" href="bagorder.html#T5" target="_self">Th5</a>, <a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:48"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E2:48"><i><font color="Green">E30</font></i></a>, <a class="ref" href="wellfnd1.html#T15">WELLFND1:15</a></i>;<br/>


</div>
