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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o1</font> being  <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/> for <font color="Olive" title="b2">o2</font> being  <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="nat_d.html#K7" title="NAT_D:func.7">-'</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>  st <font color="Olive" title="b1">o1</font> is  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a>  &amp; <font color="Olive" title="b2">o2</font> is  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a>  holds <br/> <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="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> is  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o2</font> being  <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="nat_d.html#K7" title="NAT_D:func.7">-'</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>  st <font color="Maroon" title="c3">o1</font> is  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a>  &amp; <font color="Olive" title="b1">o2</font> is  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a>  holds <br/> <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="Olive" title="b1">o2</font> is  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> </span><br/><span class="kw">let </span><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="nat_d.html#K7" title="NAT_D:func.7">-'</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  implies  <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>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:52"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c3">o1</font> is  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> 
 <span class="kw">and </span><br/><a NAME="E2:52"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c4">o2</font> is  <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<a NAME="E3:52"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T40" title="NAT_D:th.40">NAT_D:40</a></span>;<br/>
<a NAME="E4:52"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<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="pre_poly.html#K16" title="PRE_POLY:func.16">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></span>;<br/>
<a NAME="E5:52"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
<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="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></span>;<br/>
<span class="kw">set </span><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="52_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">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="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">n</font> &amp; 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> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:52_1_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E2:52_1_1"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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> )</span><br/><span class="kw">reconsider </span><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="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E2:52_1_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">CUTP1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font>;<br/><span class="kw">set </span><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/><span class="kw">set </span><font color="Maroon" title="c12">CUTQ1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font>;<br/><span class="kw">set </span><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="E4:52_1_1"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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="pre_poly.html#K16" title="PRE_POLY:func.16">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E5:52_1_1"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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="pre_poly.html#K16" title="PRE_POLY:func.16">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E6:52_1_1"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<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="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E7:52_1_1"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<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="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><span class="kw">assume </span><a NAME="E8:52_1_1"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1_1_1"/>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:52_1_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1_1_1_1_1_1"/>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:52_1_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_1_1_1_1_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:52_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1_1_1"><span class="lab"><font color="Green" title="E43">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><a NAME="E2:52_1_1_1_1_1_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E6:52_1_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E7:52_1_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">RELAT_2:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:52_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1_1_1"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:52_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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></span><br/><div class="add"><a NAME="E2:52_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E4:52_1_1"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E5:52_1_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">RELAT_2:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1_1_1_2_1_1"/>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:52_1_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_1_1_2_1_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:52_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_1_1_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><a NAME="E2:52_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E6:52_1_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E7:52_1_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">RELAT_2:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1_1_1_2_1_1_2_1_1"/>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font> or <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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="52_1_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_1_1_2_1_1_2_1_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:52_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_1_1_2_1_1_2_1_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c9">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:52_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_1_1_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:52_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:52_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:52_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="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D7" title="RELAT_2:def.7">RELAT_2:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b1">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> &amp; ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">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> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>6</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>7</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>8</sub></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> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">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> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>6</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>7</sub></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> ) )</span><br/><span class="kw">set </span><font color="Maroon" title="c7">CUTE1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c8">CUTA1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">a</font>;<br/><span class="kw">set </span><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="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/><span class="kw">set </span><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="52_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1_2_1"/>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">a</font> or <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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="52_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_2_1_1"/><span class="lab"><font color="Green" title="E36">A16</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">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></span><br/><div class="add"><a NAME="E2:52_1_2_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T17" target="_self" title="BAGORDER:th.17">Th17</a></span>;<br/><a NAME="E3:52_1_2_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:52_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="pre_poly.html#K17" title="PRE_POLY:func.17">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_2_1_1"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_2_1_2"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">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></span><br/><div class="add"><a NAME="E2:52_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="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T17" target="_self" title="BAGORDER:th.17">Th17</a></span>;<br/><a NAME="E3:52_1_2_1_2"/><span class="kw">then </span>
<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="pre_poly.html#K17" title="PRE_POLY:func.17">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:52_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="pre_poly.html#K17" title="PRE_POLY:func.17">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_2_1_2"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:52_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="pre_poly.html#K17" title="PRE_POLY:func.17">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>6</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>7</sub></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></span><br/><span class="kw">let </span><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font>, <font color="Maroon" title="c13">c</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>4</sub></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> )</span><br/><span class="kw">assume </span><a NAME="E5:52_1"/><span class="lab"><font color="Green" title="E36">A18</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>4</sub></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></span><br/><span class="kw">set </span><font color="Maroon" title="c14">CUTA1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font>;<br/><span class="kw">set </span><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/><span class="kw">set </span><font color="Maroon" title="c16">CUTB1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font>;<br/><span class="kw">set </span><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/><span class="kw">set </span><font color="Maroon" title="c18">CUTC1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font>;<br/><span class="kw">set </span><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/><span class="kw">set </span><font color="Maroon" title="c20">CUTAC1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c13">c</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">CUTBC1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c13">c</font></span>)</span>;<br/><span class="kw">set </span><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="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c13">c</font></span>)</span>;<br/><span class="kw">set </span><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="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c13">c</font></span>)</span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1_3"/>
( ( <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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 ( <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_1"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>4</sub></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></span><br/><div class="add"><a NAME="E2:52_1_3_1"/><span class="kw">then </span>
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></span>;<br/><a NAME="E3:52_1_3_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c13">c</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></span>;<br/><a NAME="E4:52_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A20</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c13">c</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="pre_poly.html#K12" title="PRE_POLY:func.12">+</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_3_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:52_1_3_1_1"/><span class="lab"><font color="Green" title="E39">A21</font></span>: 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</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> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</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> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</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> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</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> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:52_1_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</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> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T48" title="PRE_POLY:th.48">PRE_POLY:48</a></span>;<br/><span class="kw">hence </span><a NAME="E3:52_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_3_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="txt" href="#E1:52_1_3_1_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="ref" href="pre_poly.html#T48" title="PRE_POLY:th.48">PRE_POLY:48</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:52_1_3_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</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> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</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> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></span>;<br/><a NAME="E7:52_1_3_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c13">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></span>;<br/><span class="kw">hence </span><a NAME="E8:52_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="pre_poly.html#K12" title="PRE_POLY:func.12">+</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="pre_poly.html#K12" title="PRE_POLY:func.12">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_3_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_2"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Olive">b<sub>4</sub></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></span><br/><div class="add"><a NAME="E2:52_1_3_2"/><span class="kw">then </span>
<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="pre_poly.html#K12" title="PRE_POLY:func.12">+</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="pre_poly.html#K12" title="PRE_POLY:func.12">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></span>;<br/><a NAME="E3:52_1_3_2"/><span class="kw">then </span>
<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="pre_poly.html#K12" title="PRE_POLY:func.12">+</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="pre_poly.html#K12" title="PRE_POLY:func.12">+</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></span>;<br/><a NAME="E4:52_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<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="pre_poly.html#K12" title="PRE_POLY:func.12">+</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="pre_poly.html#K12" title="PRE_POLY:func.12">+</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></span>;<br/><a NAME="E5:52_1_3_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</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> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</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> <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <font color="Maroon" title="c13">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_3_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></span>;<br/><a NAME="E6:52_1_3_2"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c13">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c13">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#T18" target="_self" title="BAGORDER:th.18">Th18</a></span>;<br/><span class="kw">hence </span><a NAME="E7:52_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="pre_poly.html#K12" title="PRE_POLY:func.12">+</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="pre_poly.html#K12" title="PRE_POLY:func.12">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_3_2"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="bagorder.html#D12" target="_self" title="BAGORDER:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:52"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
