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

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

<b>set </b><font color="Maroon">ILO</font> =  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> )
 ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font>, <font color="Maroon">q</font> = <font color="Maroon">y</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1"/><i><font color="Green">E31</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:40_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; ( for <font color="Olive">i</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds <br/>(  not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> or  not <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> or  ex <font color="Olive">k</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> ) ) ) )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">s</font> =  <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">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span>;<br/><a NAME="E3:40_1_1"/><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:40_1_1"/><i><font color="Green">E48</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">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><a NAME="E5:40_1_1"/>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</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="E6:40_1_1"/><b>then </b>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</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="E7:40_1_1"/><b>then </b>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="orders_1.html#T133">ORDERS_1:133</a></i>;<br/><a NAME="E8:40_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/><a NAME="E9:40_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="triang_1.html#D2">TRIANG_1: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> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/><a NAME="E10:40_1_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:40_1_1"/><i><font color="Green">E64</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_1_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_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/><b>assume </b><a NAME="E1:40_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1_1_1_1"/>
( <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or ( <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_1_1_1_1_1"/>
<font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:40_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:40_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:40_1_1_1_1_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> &amp; <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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:40_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><b>thus </b><a NAME="E6:40_1_1_1_1_1_1"/>
<font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_1_1_1_2"/>
<font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:40_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:40_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:40_1_1_1_1_1_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> &amp; <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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:40_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><b>thus </b><a NAME="E6:40_1_1_1_1_1_2"/>
<font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:40_1_1_1_1_1_3"/>
( <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_1_1_1_1_1_3"/>
<font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:40_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:40_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>]
 <b>and </b><br/><a NAME="E14:40_1_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, );<br/></i><a NAME="E15:40_1_1"/><i><font color="Green">E84</font></i>: 
<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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">J</font> = <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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="bagorder.html#E1:40_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">J</font> = <font color="Maroon">J</font>;<br/><b>thus </b><a NAME="E1:40_1_1_2"/>
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:40_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bagorder.html#E2:40_1_1"><i><font color="Green">E36</font></i></a></i>;<br/><div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:40_1_1_2_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E2:40_1_1_2_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E3:40_1_1_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_2_1_1"/>
( not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">p</font> &amp; not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">q</font> )
 ;<br/><a NAME="E2:40_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><b>hence </b><a NAME="E3:40_1_1_2_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E5:40_1_1_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:40_1_1_2_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> &amp; <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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:40_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E8:40_1_1_2_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="bagorder.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E9:40_1_1_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:40_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="bagorder.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E10:40_1_1_2_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E11:40_1_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:40_1"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/><a NAME="E12:40_1_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E13:40_1_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E14:40_1_1_2_1"/><b>then </b>
<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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</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">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">p</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:40_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bagorder.html#E2:40_1_1"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E15:40_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E3:40_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div><a NAME="E3:40_1_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">J</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T1" target="_self">Th1</a>, <a class="txt" href="bagorder.html#E1:40_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="bagorder.html#E2:40_1_1"><i><font color="Green">E36</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:40_1_1_2"/>
<font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">J</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">J</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>thus </b><a NAME="E5:40_1_1_2"/>
for <font color="Olive">k</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">q</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:40_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:40_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:40"/>
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font> <a href="relat_2.html#R7">is_strongly_connected_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="bagorder.html#D7" target="_self">BAGORDER:def 7</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_1"/>
(  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a</font> or  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_1"><b>suppose </b></a><a NAME="E1:40_2_1_1"/>
 <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2"><b>suppose </b></a><a NAME="E1:40_2_1_2"/><i><font color="Green">E112</font></i>: 
 <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">s</font> =  <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">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span>;<br/><a NAME="E2:40_2_1_2"/><i><font color="Green">E125</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">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><a NAME="E3:40_2_1_2"/>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</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="E4:40_2_1_2"/><b>then </b>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</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="E5:40_2_1_2"/><b>then </b>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T1" target="_self">Th1</a>, <a class="ref" href="orders_1.html#T133">ORDERS_1:133</a></i>;<br/><a NAME="E6:40_2_1_2"/><b>then </b><i><font color="Green">E153</font></i>: 
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/><a NAME="E7:40_2_1_2"/><b>then </b><i><font color="Green">E154</font></i>: 
 <a href="relat_1.html#K2">rng</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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/><a NAME="E8:40_2_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E9:40_2_1_2"/><b>then </b><i><font color="Green">E155</font></i>: 
 <a href="finseq_1.html#K3">len</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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><a NAME="E10:40_2_1_2"/><b>then </b><i><font color="Green">E156</font></i>: 
<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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E1:40_2_1_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E2:40_2_1_2_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:40_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>;<br/><b>hence </b><a NAME="E4:40_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">k</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E5:40_2_1_2_1"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E6:40_2_1_2_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">j</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_2_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:40_2_1_2_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>;<br/><a NAME="E3:40_2_1_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:40_2_1_2_1_1"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> &amp; <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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E6:40_2_1_2_1_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_1_2_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> or <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_2_1_2_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_2_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_2_1_2_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</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">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:40_2_1_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/><a NAME="E3:40_2_1_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:40_2_1_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:40_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">j</font>
 <b>by </b><i>, , , , <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><a NAME="E6:40_2_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E7:40_2_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:40_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:40_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:40"/>
for <font color="Olive">a</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span>,<font color="Olive">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">c</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><b>assume </b><a NAME="E1:40_3"/><i><font color="Green">E162</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_3_1"/>
( <font color="Maroon">a</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_1"><b>suppose </b></a><a NAME="E1:40_3_1_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">a</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:40_3_1_1"/>
<font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>hence </b><a NAME="E3:40_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T1" target="_self">Th1</a>, <a class="ref" href="orders_1.html#T12">ORDERS_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_3_1_2"><b>suppose </b></a><a NAME="E1:40_3_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:40_3_1_2"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> )
 <b>and </b><br/><a NAME="E4:40_3_1_2"/><i><font color="Green">E165</font></i>: 
for <font color="Olive">k</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>thus </b><a NAME="E1:40_3_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E2:40_3_1_2_1"/><i><font color="Green">E166</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>;<br/><a NAME="E3:40_3_1_2_1"/><i><font color="Green">E167</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:40_3_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E5:40_3_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>let </b><font color="Maroon">k</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E6:40_3_1_2_1"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E7:40_3_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:40_3_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:40_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:40"/>
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">n</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#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</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#K4" target="_self">InvLexOrder</a> <font color="Maroon">n</font>
 ;<br/>


</div>
