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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">o1</font> be   <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">o2</font> be   <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c3">o1</font> is <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> &amp; <font color="Maroon" title="c4">o2</font> is <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> )
 ;<br/>

<a NAME="E2:53"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 0
 
<b>by </b><i><a class="ref" href="binarith.html#T58" title="BINARITH:th.58">BINARITH:58</a></i>;<br/>
<a NAME="E3:53"/><b>then </b><i><font color="Green" title="E33">A3</font></i>: 
( <font color="Maroon" title="c3">o1</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 0</span>)</span> &amp; <font color="Maroon" title="c4">o2</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:53"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">BO</font> =  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">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" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:53_1_1"/><i><font color="Green" title="E34">A4</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">n</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c7">y</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="txt" href="bagorder.html#E1:53_1_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>set </b><font color="Maroon" title="c10">CUTP1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font>;<br/><b>set </b><font color="Maroon" title="c11">CUTP2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font>;<br/><b>set </b><font color="Maroon" title="c12">CUTQ1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font>;<br/><b>set </b><font color="Maroon" title="c13">CUTQ2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font>;<br/><a NAME="E3:53_1_1"/><i><font color="Green" title="E35">A5</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 0</span>)</span> &amp; 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 0</span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E4:53_1_1"/><i><font color="Green" title="E36">A6</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>assume </b><a NAME="E5:53_1_1"/><i><font color="Green" title="E37">A7</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o1</font> )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E5:53_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="E38">A8</font></i>: 
0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o2</font> )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E5:53_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53_1_1_1_1"><i><font color="Green" title="E38">A8</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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:53"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="bagorder.html#E4:53_1_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53_1_1_1_1"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o1</font>
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_2"/><b>then </b><i><font color="Green" title="E38">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o1</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:53"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="bagorder.html#E3:53_1_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o2</font> )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E5:53_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:53_1_1_1_2"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o2</font>
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E39">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:53"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="bagorder.html#E4:53_1_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font> or 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:53_1_1_1_2_1_1_2"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:53_1_1_1_2"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">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" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">RELAT_2:def 7</a></i>;<br/><b>let </b><font color="Maroon" title="c6">a</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font>;<br/><b>set </b><font color="Maroon" title="c7">CUTE1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c8">CUTA1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">a</font>;<br/><b>set </b><font color="Maroon" title="c9">CUTE2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c10">CUTA2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">a</font> or 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">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" title="E34">A11</font></i>: 
0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">a</font>
 ;<br/><div class="add"><a NAME="E2:53_1_2_1_1"/>
0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 0</span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#T17" target="_self" title="BAGORDER:th.17">Th17</a></i>;<br/><a NAME="E3:53_1_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p3">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o1</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="bagorder.html#E2:53"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E4:53_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c6">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53_1_2_1_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="E34">A12</font></i>: 
0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">a</font>
 ;<br/><div class="add"><a NAME="E2:53_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#T17" target="_self" title="BAGORDER:th.17">Th17</a></i>;<br/><a NAME="E3:53_1_2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p3">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E4:53_1_2_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c6">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53_1_2_1_2"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c6">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 ;<br/><b>let </b><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font>, <font color="Maroon" title="c13">c</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font>;<br/><b>assume </b><a NAME="E5:53_1"/><i><font color="Green" title="E34">A13</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 ;<br/><b>set </b><font color="Maroon" title="c14">CUTA1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font>;<br/><b>set </b><font color="Maroon" title="c15">CUTA2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font>;<br/><b>set </b><font color="Maroon" title="c16">CUTB1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font>;<br/><b>set </b><font color="Maroon" title="c17">CUTB2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font>;<br/><b>set </b><font color="Maroon" title="c18">CUTC1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font>;<br/><b>set </b><font color="Maroon" title="c19">CUTC2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font>;<br/><b>set </b><font color="Maroon" title="c20">CUTAC1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c21">CUTBC1</font> = 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c22">CUTAC2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c23">CUTBC2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o1</font> ) or ( 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o2</font> ) )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E5:53_1"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="E35">A14</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o1</font> )
 ;<br/><div class="add"><a NAME="E2:53_1_3_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p3">(<span class="default">0,<span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p3">(<span class="default">0,<span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o1</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="bagorder.html#E2:53"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></i>;<br/><a NAME="E3:53_1_3_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p3">(<span class="default">0,<span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o1</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></i>;<br/><a NAME="E4:53_1_3_1"/><b>then </b><i><font color="Green" title="E36">A15</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span></span>)</span>,<span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o1</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></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" title="E37">A16</font></i>: 
<span class="p1">(<span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p1">(<span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p1">(<span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p2">(<span class="default">0,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span></span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <span class="p1">(<span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>;<br/><b>hence </b><a NAME="E3:53_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53_1_3_1"><i><font color="Green" title="E35">A14</font></i></a>, <a class="txt" href="bagorder.html#E1:53_1_3_1_1"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p1">(<span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></i>;<br/><a NAME="E7:53_1_3_1"/><b>then </b>
0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></i>;<br/><b>hence </b><a NAME="E8:53_1_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:53_1_3_1"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="E35">A17</font></i>: 
( 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o2</font> )
 ;<br/><div class="add"><a NAME="E2:53_1_3_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></i>;<br/><a NAME="E3:53_1_3_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">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" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></i>;<br/><a NAME="E4:53_1_3_2"/><b>then </b><i><font color="Green" title="E36">A18</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">n</font> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></i>;<br/><a NAME="E5:53_1_3_2"/>
0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p1">(<span class="default">0,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:53_1_3_2"><i><font color="Green" title="E35">A17</font></i></a>, <a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></i>;<br/><a NAME="E6:53_1_3_2"/><b>then </b>
0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> 0,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></i>;<br/><b>hence </b><a NAME="E7:53_1_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K8" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:53_1_3_2"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</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" title="BAGORDER:func.8">BlockOrder</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font> is <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a>
 <b>by </b><i><a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></i>;<br/>


</div>
