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

<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">C</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  non  <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a>  non  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">C</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">C</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">C</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  non  <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a>  non  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</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">C</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> implies (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">W</font> =  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c3">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">E</font> =  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c3">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">S</font> =  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">N</font> =  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> );<br/>
<a NAME="E1:36"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:36"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c3">C</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> or  not <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> ) )</span><br/>

<a NAME="E3:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A3</font></span>: 
 <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan9.html#D1" target="_self" title="JORDAN9:def.1">Def1</a></span>;<br/>
<a NAME="E4:36"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></span>;<br/>
<a NAME="E5:36"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_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="E1:36_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></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">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:36_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:36_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E37">A6</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="36_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></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">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:36_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/><a NAME="E3:36_1_1_2"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E4:36_1_1_2"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="goboard1.html#T38" title="GOBOARD1:th.38">GOBOARD1:38</a></span>;<br/><a NAME="E5:36_1_1_2"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="goboard1.html#T38" title="GOBOARD1:th.38">GOBOARD1:38</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:36_1_1_2"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">and </span><br/><a NAME="E8:36_1_1_2"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:36_1_1_2"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
( <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E10:36_1_1_2"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:36_1_1_2"/>
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="jordan9.html#D1" target="_self" title="JORDAN9:def.1">Def1</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> &amp; <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E12:36_1_1_2"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">and </span><br/><a NAME="E13:36_1_1_2"/><span class="lab"><font color="Green" title="E48">A17</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><a NAME="E14:36_1_1_2"/><span class="lab"><font color="Green" title="E49">A18</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_1_2"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E15:36_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/><a NAME="E16:36_1_1_2"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T104" title="NEWTON:th.104">NEWTON:104</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_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="E1:36_1_1_2_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E3:36_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E13:36_1_1_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:36_1_1_2_1_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</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="36_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E52">A22</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:36_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A23</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E13:36_1_1_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E9:36_1_1_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="ref" href="finseq_5.html#T84" title="FINSEQ_5:th.84">FINSEQ_5:84</a></span>;<br/><a NAME="E5:36_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A24</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></span>;<br/><a NAME="E6:36_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36_1_1_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A25</font></span>: 
<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">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</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>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E8:36_1_1_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E3:36_1_1_2"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>;<br/><a NAME="E8:36_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E56">A26</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36_1_1_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E9:36_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A27</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</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>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E7:36_1_1_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E3:36_1_1_2"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>;<br/><a NAME="E10:36_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E58">A28</font></span>: 
( <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E11:36_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A29</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E5:36_1_1_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E13:36_1_1_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:36_1_1_2_1_1_2_1"/>
 <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#T12" title="JORDAN8:th.12">JORDAN8:12</a></span>;<br/><a NAME="E2:36_1_1_2_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E3:36_1_1_2_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:36_1_1_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A30</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">assume </span><a NAME="E5:36_1_1_2_1_1_2_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:36_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E61">A31</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:36_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E62">A32</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">p</font> = <font color="Maroon" title="c13">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E61">A31</font></span></a></span>;<br/><a NAME="E10:36_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E63">A33</font></span>: 
<font color="Maroon" title="c14">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E62">A32</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/><a NAME="E11:36_1_1_2_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</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>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E3:36_1_1_2"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>;<br/><a NAME="E12:36_1_1_2_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p5">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p5">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></span>;<br/><a NAME="E13:36_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A34</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/><a NAME="E14:36_1_1_2_1_1_2_1"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E7:36_1_1_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="ref" href="gobrd11.html#T31" title="GOBRD11:th.31">GOBRD11:31</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">r</font>, <font color="Maroon" title="c16">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E16:36_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E65">A35</font></span>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c15">r</font>,<font color="Maroon" title="c16">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E17:36_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">r</font>
 <span class="kw">and </span><br/><a NAME="E18:36_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E19:36_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E66">A36</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E61">A31</font></span></a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E35">A4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:36_1_1_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
;<br/><a NAME="E21:36_1_1_2_1_1_2_1"/><span class="kw">then </span>
 <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c3">C</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:36_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E66">A36</font></span></a>, <a class="txt" href="#E13:36_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E64">A34</font></span></a>, <a class="txt" href="#E4:36_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E22:36_1_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:36_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E65">A35</font></span></a>, <a class="txt" href="#E10:36_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E63">A33</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</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><span class="kw">hence </span><a NAME="E13:36_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E14:36_1_1_2_1_1_2"/>
(  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E5:36_1_1_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E10:36_1_1_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E13:36_1_1_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="txt" href="#E10:36_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E58">A28</font></span></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a>, <a class="ref" href="sprect_1.html#T13" title="SPRECT_1:th.13">SPRECT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E15:36_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</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="36_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E52">A37</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A38</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E3:36_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E54">A39</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_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="E1:36_1_1_2_1_1_3_1_1"/>
( <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> or <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E55">A40</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E56">A41</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T18" title="FINSEQ_5:th.18">FINSEQ_5:18</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a></span>;<br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A42</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>;<br/><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A43</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a></span>;<br/><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A44</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">i1</font>, <font color="Maroon" title="c14">j1</font>, <font color="Maroon" title="c15">i2</font>, <font color="Maroon" title="c16">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A45</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</font> )
 <span class="kw">and </span><br/><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A46</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:36_1_1_2_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A47</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font>
 <span class="kw">and </span><br/><a NAME="E10:36_1_1_2_1_1_3_1_1_1_1"/>
( ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E1:36_1_1_2"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E11:36_1_1_2_1_1_3_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E12:36_1_1_2_1_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><a NAME="E13:36_1_1_2_1_1_3_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/><a NAME="E14:36_1_1_2_1_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A49</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_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="E1:36_1_1_2_1_1_3_1_1_1_1_1"/>
( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> ) or (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> ) or  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E65">A50</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A51</font></span>: 
 <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="ref" href="jordan9.html#D1" target="_self" title="JORDAN9:def.1">Def1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_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="E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> &amp; <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="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <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="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E66">A51</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E67">A52</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E68">A53</font></span>: 
( <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="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c16">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E67">A52</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A53</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E67">A54</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E68">A55</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E14:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E14:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E67">A56</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E68">A57</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E67">A56</font></span></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A57</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E67">A56</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A57</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E67">A58</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E68">A59</font></span>: 
( <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="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E67">A58</font></span></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E12:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A50</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E68">A59</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E67">A58</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E12:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E68">A59</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="E4:36_1_1_2_1_1_3_1_1_1_1_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</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="36_1_1_2_1_1_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E65">A60</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A61</font></span>: 
 <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="ref" href="jordan9.html#D1" target="_self" title="JORDAN9:def.1">Def1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_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="E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c13">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> &amp; <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="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> &amp; <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="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E66">A61</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E67">A62</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E68">A63</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A62</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A60</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A62</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A63</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c16">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A62</font></span></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A60</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A62</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A63</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E67">A64</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E68">A65</font></span>: 
( <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="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A64</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A60</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A65</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A64</font></span></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A60</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A65</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E67">A66</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E68">A67</font></span>: 
( <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="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E67">A66</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E12:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A60</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E67">A66</font></span></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E12:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A60</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E67">A68</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E68">A69</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E67">A68</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E14:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A60</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E68">A69</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E67">A68</font></span></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E14:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A60</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E68">A69</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="E4:36_1_1_2_1_1_3_1_1_1_1_1_2"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</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="36_1_1_2_1_1_3_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E65">A70</font></span>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A71</font></span>: 
 <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="ref" href="jordan9.html#D1" target="_self" title="JORDAN9:def.1">Def1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_1_1_1_3_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="E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> &amp; <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="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <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="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E66">A71</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E67">A72</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E68">A73</font></span>: 
( <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="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/>
( <font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E67">A72</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E12:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E67">A72</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E68">A73</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E67">A72</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E12:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E63">A48</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E65">A70</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E67">A72</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E68">A73</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E67">A74</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E68">A75</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
( <font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E67">A74</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E67">A74</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E68">A75</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E67">A74</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E65">A70</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E68">A75</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E67">A76</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E68">A77</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E67">A76</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E14:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E68">A77</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E67">A76</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E14:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E64">A49</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E65">A70</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E68">A77</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E67">A78</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E68">A79</font></span>: 
( <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="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E3:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E67">A78</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>;<br/><a NAME="E4:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E68">A79</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font></span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E67">A78</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>;<br/><a NAME="E7:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A46</font></span></a>, <a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A47</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E65">A70</font></span></a>, <a class="txt" href="#E2:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E68">A79</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</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="E4:36_1_1_2_1_1_3_1_1_1_1_1_3"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</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="E4:36_1_1_2_1_1_3_1_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</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="36_1_1_2_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_3_1_1_2"/>
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36_1_1_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:36_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A80</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E15:36_1_1_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">i1</font>, <font color="Maroon" title="c14">j1</font>, <font color="Maroon" title="c15">i2</font>, <font color="Maroon" title="c16">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:36_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E56">A81</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:36_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E57">A82</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</font>
 <span class="kw">and </span><br/><a NAME="E7:36_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E58">A83</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</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="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:36_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E59">A84</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font>
 <span class="kw">and </span><br/><a NAME="E9:36_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E60">A85</font></span>: 
( ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1_1_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/><a NAME="E10:36_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E61">A86</font></span>: 
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A80</font></span></a></span>;<br/><a NAME="E11:36_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E62">A87</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:36_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="ref" href="finseq_5.html#T85" title="FINSEQ_5:th.85">FINSEQ_5:85</a></span>;<br/><a NAME="E12:36_1_1_2_1_1_3_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E13:36_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A80</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E14:36_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A88</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A84</font></span></a>, <a class="txt" href="#E11:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E62">A87</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><a NAME="E15:36_1_1_2_1_1_3_1_1_2"/>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A80</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E16:36_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E17:36_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A89</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A82</font></span></a>, <a class="txt" href="#E11:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E62">A87</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_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="E1:36_1_1_2_1_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> ) or ( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A85</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_1_1_3_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E65">A90</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_3_1_1_2_1_1_1"/><span class="kw">then </span>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c14">j1</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1_1_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A80</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E6:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A82</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A84</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:36_1_1_2_1_1_3_1_1_2_1_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_1_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E13:36_1_1_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="txt" href="#E10:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E61">A86</font></span></a>, <a class="txt" href="#E17:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E64">A89</font></span></a>, <a class="txt" href="#E14:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E63">A88</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A90</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</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="36_1_1_2_1_1_3_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E65">A91</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_3_1_1_2_1_1_2"/><span class="kw">then </span>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i1</font>,<font color="Maroon" title="c14">j1</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c13">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1_1_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A80</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E6:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A82</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A84</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><span class="kw">hence </span><a NAME="E3:36_1_1_2_1_1_3_1_1_2_1_1_2"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_1_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E13:36_1_1_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="txt" href="#E10:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E61">A86</font></span></a>, <a class="txt" href="#E17:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E64">A89</font></span></a>, <a class="txt" href="#E14:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E63">A88</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A91</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</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="36_1_1_2_1_1_3_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_3_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E65">A92</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_3_1_1_2_1_1_3"/><span class="kw">then </span>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1_1_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A80</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E6:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A82</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A84</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><span class="kw">hence </span><a NAME="E3:36_1_1_2_1_1_3_1_1_2_1_1_3"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_1_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E13:36_1_1_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="txt" href="#E10:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E61">A86</font></span></a>, <a class="txt" href="#E17:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E64">A89</font></span></a>, <a class="txt" href="#E14:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E63">A88</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E65">A92</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</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="36_1_1_2_1_1_3_1_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:36_1_1_2_1_1_3_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E65">A93</font></span>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i2</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )</span><br/><div class="add"><a NAME="E2:36_1_1_2_1_1_3_1_1_2_1_1_4"/><span class="kw">then </span>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i2</font>,<font color="Maroon" title="c16">j2</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1_1_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E3:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A80</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E6:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A82</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="txt" href="#E8:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A84</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/><span class="kw">hence </span><a NAME="E3:36_1_1_2_1_1_3_1_1_2_1_1_4"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_1_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E12:36_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E13:36_1_1_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E5:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A81</font></span></a>, <a class="txt" href="#E7:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A83</font></span></a>, <a class="txt" href="#E10:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E61">A86</font></span></a>, <a class="txt" href="#E17:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E64">A89</font></span></a>, <a class="txt" href="#E14:36_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E63">A88</font></span></a>, <a class="txt" href="#E1:36_1_1_2_1_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E65">A93</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</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="E19:36_1_1_2_1_1_3_1_1_2"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</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="E5:36_1_1_2_1_1_3"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</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="E18:36_1_1_2"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c12">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</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>
<a NAME="E6:36"/><span class="lab"><font color="Green" title="E37">A94</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></span>;<br/>
<a NAME="E7:36"/><span class="lab"><font color="Green" title="E38">A95</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></span>;<br/>
<a NAME="E8:36"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E7:36"><span class="lab"><font color="Green" title="E38">A95</font></span></a>, <a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E36">A5</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:36"/>
(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> or  not <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">C</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E37">A94</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>
