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

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="bagorder.html#NM1">TermOrder</a> of <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="bagorder.html#NM1">TermOrder</a> of <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>;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">o1</font> is <a href="bagorder.html#V2" target="_self">admissible</a> &amp; <font color="Maroon">o2</font> is <a href="bagorder.html#V2" target="_self">admissible</a> )
 ;<br/>

<a NAME="E2:53"/><i><font color="Green">E31</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/>
<a NAME="E3:53"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">o1</font> <a href="relat_2.html#R7">is_strongly_connected_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; <font color="Maroon">o2</font> <a href="relat_2.html#R7">is_strongly_connected_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="txt" href="bagorder.html#E2:53"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">BO</font> =  <a href="bagorder.html#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_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> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1_1"/><i><font color="Green">E38</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/><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="E3:53_1_1"/><i><font color="Green">E48</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="E4:53_1_1"/><i><font color="Green">E49</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/><b>assume </b><a NAME="E5:53_1_1"/><i><font color="Green">E50</font></i>: 
not <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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_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">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> or  not <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> )
 </span><b>by </b><i>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_1"/><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="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>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_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="E1:53_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">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> or  not <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>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_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">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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1_1_1_1_1"/>
<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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53_1_1"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_1_1_2"/>
not <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"><a NAME="E2:53_1_1_1_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"><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#E3:53"><i><font color="Green">E36</font></i></a>, <a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>;<br/><b>hence </b><a NAME="E3:53_1_1_1_1_1_1_2"/>
<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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</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="E3:53_1_1_1_1"/>
<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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_2"/>
not <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"><a NAME="E2:53_1_1_1_2"/><b>then </b><i><font color="Green">E64</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">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>
 <b>by </b><i>, , <a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_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">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> or  not <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>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_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">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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1_2_1_1_1"/>
<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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_2_1_1_2"/>
not <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"><a NAME="E2:53_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<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#E3:53"><i><font color="Green">E36</font></i></a>, <a class="ref" href="relat_2.html#D7">RELAT_2:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_1_2_1_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">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> 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="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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_2_1_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">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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1_2_1_1_2_1_1_1"/>
<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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 <b>by </b><i>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_2_1_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">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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1_2_1_1_2_1_1_2"/>
<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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</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="E4:53_1_1_1_2_1_1_2"/>
<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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:53_1_1_1_2"/>
<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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:53_1"/>
 <a href="bagorder.html#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</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>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">CUTE1</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> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span>;<br/><b>set </b><font color="Maroon">CUTA1</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">a</font>;<br/><b>set </b><font color="Maroon">CUTE2</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> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span>;<br/><b>set </b><font color="Maroon">CUTA2</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">a</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_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> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <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">a</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> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <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">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2_1_1"><b>suppose </b></a><a NAME="E1:53_1_2_1_1"/><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> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <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">a</font>
 ;<br/><div class="add"><a NAME="E2:53_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> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</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/>;<br/><a NAME="E3:53_1_2_1_1"/><b>then </b>
<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> <span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span></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">a</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#E2:53"><i><font color="Green">E31</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E4:53_1_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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 <b>by </b><i>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2_1_2"><b>suppose </b></a><a NAME="E1:53_1_2_1_2"/><i><font color="Green">E69</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> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <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">a</font>
 ;<br/><div class="add"><a NAME="E2:53_1_2_1_2"/>
<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> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</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/>;<br/><a NAME="E3:53_1_2_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> <span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span></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">a</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#E2:53"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:53_1_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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 <b>by </b><i>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</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="E4:53_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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 ;<br/><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="E5:53_1"/><i><font color="Green">E70</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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 ;<br/><b>set </b><font color="Maroon">CUTA1</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">a</font>;<br/><b>set </b><font color="Maroon">CUTA2</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">a</font>;<br/><b>set </b><font color="Maroon">CUTB1</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">b</font>;<br/><b>set </b><font color="Maroon">CUTB2</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">b</font>;<br/><b>set </b><font color="Maroon">CUTC1</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">c</font>;<br/><b>set </b><font color="Maroon">CUTC2</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">c</font>;<br/><b>set </b><font color="Maroon">CUTAC1</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> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span>;<br/><b>set </b><font color="Maroon">CUTBC1</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> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span>;<br/><b>set </b><font color="Maroon">CUTAC2</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> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span>;<br/><b>set </b><font color="Maroon">CUTBC2</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> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_3"/>
( ( 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">a</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">b</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">a</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">b</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">a</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">b</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">a</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">b</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>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3_1"><b>suppose </b></a><a NAME="E1:53_1_3_1"/><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">a</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">b</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">a</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">b</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="E2:53_1_3_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">0,<span class="p4">(<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">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p3">(<span class="default">0,<span class="p4">(<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">c</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">0,<span class="p4">(<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">b</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p3">(<span class="default">0,<span class="p4">(<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">c</font></span>)</span></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#E2:53"><i><font color="Green">E31</font></i></a>, , </i>;<br/><a NAME="E3:53_1_3_1"/><b>then </b>
<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> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">0,<span class="p4">(<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">b</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p3">(<span class="default">0,<span class="p4">(<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">c</font></span>)</span></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/>;<br/><a NAME="E4:53_1_3_1"/><b>then </b><i><font color="Green">E84</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> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></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/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1_3_1_1"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default">0,<span class="p2">(<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">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default">0,<span class="p2">(<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">c</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default">0,<span class="p2">(<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">b</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default">0,<span class="p2">(<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">c</font></span>)</span>
 ;<br/><a NAME="E2:53_1_3_1_1"/>
<span class="p1">(<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">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <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">c</font></span>)</span></span>)</span> <a href="polynom1.html#K10">-'</a> <span class="p1">(<span class="default">0,<span class="p2">(<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">c</font></span>)</span> <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">a</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>;<br/><b>hence </b><a NAME="E3:53_1_3_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>;<br/></div><b>end;</b></div><a NAME="E6:53_1_3_1"/><b>then </b>
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> <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="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default">0,<span class="p2">(<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">b</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default">0,<span class="p2">(<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">c</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:53_1_3_1"/><b>then </b>
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> <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="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> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:53_1_3_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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:53"><i><font color="Green">E36</font></i></a>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3_2"><b>suppose </b></a><a NAME="E1:53_1_3_2"/><i><font color="Green">E86</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">a</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">b</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">a</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">b</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="E2:53_1_3_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"><span class="p4">(<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">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<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">c</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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">b</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<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">c</font></span>)</span></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#E2:53"><i><font color="Green">E31</font></i></a>, </i>;<br/><a NAME="E3:53_1_3_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> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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">b</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<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">c</font></span>)</span></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/><a NAME="E4:53_1_3_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<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> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span></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/><a NAME="E5:53_1_3_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> <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="pboole.html#R6">=</a> <span class="p1">(<span class="default">0,<span class="p2">(<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">b</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default">0,<span class="p2">(<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">c</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:53_1_3_2"/><b>then </b>
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> <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="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> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:53_1_3_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#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:53"><i><font color="Green">E36</font></i></a>, <a class="txt" href="bagorder.html#E1:53"><i><font color="Green">E30</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="E5:53"/>
 <a href="bagorder.html#K8" target="_self">BlockOrder</a> <font color="Maroon">i</font>,<font color="Maroon">n</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font> is <a href="bagorder.html#V2" target="_self">admissible</a>
 <b>by </b><i/>;<br/>


</div>
