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

<a NAME="E1:52_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 0
 
<b>by </b><i><a class="ref" href="jordan3.html#T2">JORDAN3:2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; ( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Olive">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Olive">p</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Olive">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Olive">p</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Olive">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Olive">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Olive">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) ) );<br/>
<b>consider </b><font color="Maroon">BO</font> being    <a href="relset_1.html#M2">Relation</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>, <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:52_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="relset_1.html#S1">RELSET_1:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_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="E1:52_1_1_1"/><i><font color="Green">E36</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>
 ;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</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/><a NAME="E3:52_1_1_1"/><i><font color="Green">E38</font></i>: 
0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font>
 ;<br/><a NAME="E4:52_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E5:52_1_1_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"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T12">ORDERS_1:12</a></i>;<br/><b>hence </b><a NAME="E6:52_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:52_1_1"><i><font color="Green">E31</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E5:52_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">BO</font> <a href="relat_2.html#R1">is_reflexive_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#D1">RELAT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2"><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><b>that </b><br/><a NAME="E1:52_1_1_2"/>
( <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> )
 <b>and </b><br/><a NAME="E2:52_1_1_2"/><i><font color="Green">E49</font></i>: 
( <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> <font color="Maroon">BO</font> &amp; <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> <font color="Maroon">BO</font> )
 ;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">q</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E4:52_1_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>and </b><br/><a NAME="E5:52_1_1_2"/><i><font color="Green">E62</font></i>: 
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E3:52_1_1"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">q'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">p'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E7:52_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> )
 <b>and </b><br/><a NAME="E8:52_1_1_2"/><i><font color="Green">E67</font></i>: 
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q'</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E3:52_1_1"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">CUTP1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font>;<br/><b>set </b><font color="Maroon">CUTP2</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font>;<br/><b>set </b><font color="Maroon">CUTQ1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font>;<br/><b>set </b><font color="Maroon">CUTQ2</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font>;<br/><a NAME="E9:52_1_1_2"/><i><font color="Green">E68</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 0</span>)</span> &amp; 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 0</span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E10:52_1_1_2"/><i><font color="Green">E69</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_2_1"/>
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_2_1_1"/><i><font color="Green">E70</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2_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="E1:52_1_1_2_1_1_1_1"/>
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_2_1_1_1_1_1"/>
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_2_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:52_1_1"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="orders_1.html#T13">ORDERS_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_2_1_1_1_1_2"/>
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_2_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</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="E3:52_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_2_1_2"/><i><font color="Green">E71</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_2_1_2_1_1"/>
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_2_1_2_1_1_1"/>
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_2_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_2_1_2_1_1_2"/>
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> )
 ;<br/><div class="add"><a NAME="E2:52_1_1_2_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="orders_1.html#T13">ORDERS_1:13</a></i>;<br/><b>hence </b><a NAME="E3:52_1_1_2_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:52_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:52_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">BO</font> <a href="relat_2.html#R4">is_antisymmetric_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#D4">RELAT_2:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_3"><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> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:52_1_1_3"/><i><font color="Green">E85</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> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E2:52_1_1_3"/><i><font color="Green">E86</font></i>: 
( <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> <font color="Maroon">BO</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font> )
 ;<br/><b>consider </b><font color="Maroon">x'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">y'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E4:52_1_1_3"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>and </b><br/><a NAME="E5:52_1_1_3"/><i><font color="Green">E88</font></i>: 
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E3:52_1_1"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">y''</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">z'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E7:52_1_1_3"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">y''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 <b>and </b><br/><a NAME="E8:52_1_1_3"/><i><font color="Green">E111</font></i>: 
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y''</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y''</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y''</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y''</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E3:52_1_1"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">CUTX1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font>;<br/><b>set </b><font color="Maroon">CUTX2</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font>;<br/><b>set </b><font color="Maroon">CUTY1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font>;<br/><b>set </b><font color="Maroon">CUTY2</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font>;<br/><b>set </b><font color="Maroon">CUTZ1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font>;<br/><b>set </b><font color="Maroon">CUTZ2</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font>;<br/><a NAME="E9:52_1_1_3"/><i><font color="Green">E112</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 0</span>)</span> &amp; 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 0</span>)</span> &amp; 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 0</span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E10:52_1_1_3"/><i><font color="Green">E125</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_3_1"/>
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_3_1_1"/><i><font color="Green">E153</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_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="E1:52_1_1_3_1_1_1_1"/>
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_3_1_1_1_1_1"/><i><font color="Green">E154</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> )
 ;<br/><div class="add"><a NAME="E2:52_1_1_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E155</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:52_1_1"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="orders_1.html#T14">ORDERS_1:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_3_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_1_1_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="E1:52_1_1_3_1_1_1_1_1_1_1"/>
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> or 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_3_1_1_1_1_1_1_1_1"/>
0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_3_1_1_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:52_1_1"><i><font color="Green">E31</font></i></a>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_3_1_1_1_1_1_1_1_2"/>
0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_3_1_1_1_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:52_1_1"><i><font color="Green">E30</font></i></a>, , , , <a class="ref" href="orders_1.html#T13">ORDERS_1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:52_1_1_3_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_3_1_1_1_1_2"/>
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_3_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:52_1_1"><i><font color="Green">E31</font></i></a>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:52_1_1_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_3_1_2"/><i><font color="Green">E156</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_3_1_2_1_1"/>
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_3_1_2_1_1_1"/>
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_1_3_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:52_1_1"><i><font color="Green">E31</font></i></a>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_3_1_2_1_1_2"/><i><font color="Green">E157</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">y'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> )
 ;<br/><div class="add"><a NAME="E2:52_1_1_3_1_2_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"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">z'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, , <a class="ref" href="orders_1.html#T14">ORDERS_1:14</a></i>;<br/><b>hence </b><a NAME="E3:52_1_1_3_1_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:52_1_1"><i><font color="Green">E31</font></i></a>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:52_1_1_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BO</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E9:52_1_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">BO</font> <a href="relat_2.html#R8">is_transitive_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#D8">RELAT_2:def 8</a></i>;<br/>
<a NAME="E10:52_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">BO</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">BO</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">BO</font> = <font color="Maroon">BO</font> as   <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font> <b>by </b><i>, , , <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon">BO</font>
;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">q</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:52_1_1_4"/>
<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> <font color="Maroon">BO</font>
 ;<br/><b>then consider </b><font color="Maroon">p'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">q'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:52_1_1_4"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">p'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">q</font> &amp; ( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p'</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p'</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) ) )
 <b>by </b><i><a class="txt" href="bagorder.html#E3:52_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>thus </b><a NAME="E4:52_1_1_4"/>
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E13:52_1_1"/><i><font color="Green">E160</font></i>: 
( ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font> <a href="pboole.html#R6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="bagorder.html#K1" target="_self">-cut</a> <font color="Maroon">q</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">o2</font> ) )
 ;<br/>

<a NAME="E14:52_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> &amp; <font color="Maroon">q</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="E15:52_1_1"/>
<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> <font color="Maroon">BO</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:52_1_1"><i><font color="Green">E31</font></i></a>, </i>;<br/>


</div>
