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

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font>)</span>)</span> holds <br/>(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font>)</span>)</span>,<font color="Olive" title="b2">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font>)</span>)</span>,<font color="Olive" title="b2">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> implies (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">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="c1">C</font>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c1">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="c1">C</font> );<br/>
<a NAME="E1:9"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:9"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c3">k</font> or  not <font color="Maroon" title="c3">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> or (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> ) )</span><br/>

<a NAME="E3:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A3</font></span>: 
 <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></span>;<br/>
<a NAME="E4:9"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c1">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"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></span>;<br/>
<a NAME="E5:9"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c1">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"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></span>;<br/>
<a NAME="E6:9"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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="9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span> holds <br/>(  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:9_1"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="goboard1.html#T22" title="GOBOARD1:th.22">GOBOARD1:22</a></span>;<br/>
<a NAME="E3:9_1"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="goboard1.html#T22" title="GOBOARD1:th.22">GOBOARD1:22</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_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="E1:9_1_1"/>
( <font color="Maroon" title="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> or <font color="Maroon" title="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="9_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_1"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<font color="Maroon" title="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E3:9_1_1_1"/>
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></span>;<br/><span class="kw">hence </span><a NAME="E4:9_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="#E1:9_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E2:9_1_1_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
<font color="Maroon" title="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c7">m</font> &amp; <font color="Maroon" title="c7">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies (  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:9_1_1_2"/><span class="lab"><font color="Green" title="E16">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
<font color="Maroon" title="c7">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><a NAME="E4:9_1_1_2"/><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<font color="Maroon" title="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">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="E5:9_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A16</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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c6">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#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/><a NAME="E6:9_1_1_2"/><span class="lab"><font color="Green" title="E20">A17</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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_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="E1:9_1_1_2_1_1"/>
( <font color="Maroon" title="c6">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="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">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#T25" title="NAT_1:th.25">NAT_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E21">A18</font></span>: 
<font color="Maroon" title="c6">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E2:9_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">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="E3:9_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E21">A18</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="E4:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E21">A19</font></span>: 
<font color="Maroon" title="c6">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">j</font> =  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>);<br/><span class="kw">set </span><font color="Maroon" title="c9">i</font> =  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>);<br/><a NAME="E2:9_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E22">A20</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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_1.html#K10" title="FINSEQ_1:func.10">&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="c1">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"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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="c1">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"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E21">A19</font></span></a>, <a class="ref" href="finseq_5.html#T81" title="FINSEQ_5:th.81">FINSEQ_5:81</a></span>;<br/><a NAME="E3:9_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A21</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><a NAME="E4:9_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="c7">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="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E5:9_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A22</font></span>: 
<font color="Maroon" title="c7">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="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E21">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:9_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E25">A23</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="jordan11.html#T9" title="JORDAN11:th.9">JORDAN11:9</a></span>;<br/><a NAME="E7:9_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E26">A24</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="jordan11.html#T8" title="JORDAN11:th.8">JORDAN11:8</a></span>;<br/><a NAME="E8:9_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/><a NAME="E9:9_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A25</font></span>: 
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <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> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E10:9_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E28">A26</font></span>: 
 <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E11:9_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E29">A27</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E22">A20</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><a NAME="E12:9_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"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">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="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E25">A23</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E24">A22</font></span></a>, <a class="txt" href="#E10:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="txt" href="#E9:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E13:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="jordan11.html#T11" title="JORDAN11:th.11">JORDAN11:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E14:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">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="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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="#E2:9_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="txt" href="#E11:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E25">A23</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E24">A22</font></span></a>, <a class="txt" href="#E10:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="txt" href="#E9:9_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E15:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="jordan11.html#T10" title="JORDAN11:th.10">JORDAN11:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E21">A28</font></span>: 
<font color="Maroon" title="c6">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E2:9_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E22">A29</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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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#T16" title="FINSEQ_5:th.16">FINSEQ_5:16</a></span>;<br/><a NAME="E3:9_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E23">A30</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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><a NAME="E4:9_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E24">A31</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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E5:9_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A32</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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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="9_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="E1:9_1_1_2_1_1_3_1_1"/>
( <font color="Maroon" title="c7">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> or <font color="Maroon" title="c7">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="9_1_1_2_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E26">A33</font></span>: 
<font color="Maroon" title="c7">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E2:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a></span>;<br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A34</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">i1</font>, <font color="Maroon" title="c9">j1</font>, <font color="Maroon" title="c10">i2</font>, <font color="Maroon" title="c11">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:9_1_1_2_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E28">A35</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:9_1_1_2_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E29">A36</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c8">i1</font>,<font color="Maroon" title="c9">j1</font>)
 <span class="kw">and </span><br/><a NAME="E7:9_1_1_2_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E30">A37</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:9_1_1_2_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E31">A38</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c10">i2</font>,<font color="Maroon" title="c11">j2</font>)
 <span class="kw">and </span><br/><a NAME="E9:9_1_1_2_1_1_3_1_1_1"/>
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">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="c11">j2</font> ) or ( <font color="Maroon" title="c8">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="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E10:9_1_1_2_1_1_3_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E11:9_1_1_2_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A39</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">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="c10">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E12:9_1_1_2_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E33">A40</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E13:9_1_1_2_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A41</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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="c11">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E14:9_1_1_2_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a></span>;<br/><a NAME="E15:9_1_1_2_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A42</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_1_1_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_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="E1:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_1_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A43</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A44</font></span>: 
 <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E25">A32</font></span></a>, <a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">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="c11">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>) ) or ( <font color="Maroon" title="c8">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="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A44</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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A45</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">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="c11">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A46</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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A47</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</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="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A43</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A45</font></span></a>, <a class="ref" href="gobrd13.html#T34" title="GOBRD13:th.34">GOBRD13:34</a></span>;<br/><a NAME="E5:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E11:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A45</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A47</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="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E7:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c9">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E8:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E27">A34</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A45</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>;<br/><a NAME="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E11:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A45</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A47</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A48</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="E10:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A49</font></span>: 
( <font color="Maroon" title="c8">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="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A50</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A51</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
 <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A43</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A49</font></span></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></span>;<br/><a NAME="E5:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A50</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A51</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="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E7:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A52</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E8:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
 <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c11">j2</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E27">A34</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A49</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><a NAME="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A49</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A50</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A51</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A52</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>;<br/><span class="kw">hence </span><a NAME="E10:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E38">A53</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E39">A54</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E40">A55</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">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="c11">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A43</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E38">A53</font></span></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></span>;<br/><a NAME="E5:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E13:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E39">A54</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E40">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="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E7:9_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E27">A34</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E38">A53</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><a NAME="E8:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E13:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E39">A54</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E40">A55</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="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E38">A56</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E39">A57</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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E40">A58</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A43</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E38">A56</font></span></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></span>;<br/><a NAME="E5:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E38">A56</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E39">A57</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E40">A58</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="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E7:9_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c11">j2</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E27">A34</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E38">A56</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><a NAME="E8:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E38">A56</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E39">A57</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E40">A58</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="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_1_1_2_1_1_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A59</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A60</font></span>: 
 <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E25">A32</font></span></a>, <a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">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="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c8">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="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>) ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>) ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">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="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A60</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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E38">A61</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">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="c11">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E39">A62</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E40">A63</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A59</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>;<br/><a NAME="E5:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E39">A62</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E40">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="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E7:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A59</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="ref" href="gobrd13.html#T34" title="GOBRD13:th.34">GOBRD13:34</a></span>;<br/><a NAME="E8:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E39">A62</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E40">A63</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>;<br/><span class="kw">hence </span><a NAME="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E38">A64</font></span>: 
( <font color="Maroon" title="c8">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="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E39">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E40">A66</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A59</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E38">A64</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>;<br/><a NAME="E5:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E38">A64</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A65</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A66</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="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E7:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A59</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E38">A64</font></span></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></span>;<br/><a NAME="E8:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E38">A64</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A65</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A66</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="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E38">A67</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E39">A68</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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E40">A69</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_1_1_2_1_1_3_1_1_1_1_1_2_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A59</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E38">A67</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>;<br/><a NAME="E5:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E11:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E39">A68</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E40">A69</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="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E7:9_1_1_2_1_1_3_1_1_1_1_1_2_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">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="c11">j2</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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A59</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E38">A67</font></span></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></span>;<br/><a NAME="E8:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E11:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E39">A68</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E40">A69</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="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E38">A70</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E39">A71</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E40">A72</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E41">A73</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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="c11">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E33">A40</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E5:9_1_1_2_1_1_3_1_1_1_1_1_2_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">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="c11">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A59</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E38">A70</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>;<br/><a NAME="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E39">A71</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E40">A72</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E41">A73</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="E7:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E8:9_1_1_2_1_1_3_1_1_1_1_1_2_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A59</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E38">A70</font></span></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></span>;<br/><a NAME="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E13:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E39">A71</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E40">A72</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="E10:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_1_1_2_1_1_3_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_3_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E36">A74</font></span>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">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="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A75</font></span>: 
 <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E25">A32</font></span></a>, <a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">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="c11">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>) ) or ( <font color="Maroon" title="c8">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="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E37">A75</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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E38">A76</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">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="c11">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E39">A77</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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E40">A78</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E41">A79</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c9">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E5:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E15:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E35">A42</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E38">A76</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>;<br/><a NAME="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E38">A76</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A77</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E41">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="E7:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E8:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E38">A76</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>;<br/><a NAME="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A77</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E40">A78</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="E10:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E38">A80</font></span>: 
( <font color="Maroon" title="c8">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="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E39">A81</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E40">A82</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E41">A83</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E5:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
 <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E15:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E35">A42</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E38">A80</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>;<br/><a NAME="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E13:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E38">A80</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A81</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E40">A82</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E41">A83</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="E7:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E8:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
 <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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E38">A80</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>;<br/><a NAME="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E13:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A81</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E40">A82</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="E10:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E38">A84</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E39">A85</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E40">A86</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E15:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E35">A42</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E38">A84</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>;<br/><a NAME="E5:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A85</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E40">A86</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="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E7:9_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E38">A84</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>;<br/><a NAME="E8:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A85</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E40">A86</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>;<br/><span class="kw">hence </span><a NAME="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_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="E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E38">A87</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E39">A88</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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E40">A89</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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"><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"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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> 2</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="c1">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E4:9_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E15:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E35">A42</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E38">A87</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>;<br/><a NAME="E5:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E11:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E39">A88</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E40">A89</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="E6:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font></span><br/><a NAME="E7:9_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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">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="c11">j2</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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E36">A74</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E38">A87</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>;<br/><a NAME="E8:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E21">A28</font></span></a>, <a class="txt" href="#E4:9_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E11:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E39">A88</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E40">A89</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="E9:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="ref" href="gobrd13.html#T31" title="GOBRD13:th.31">GOBRD13:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E17:9_1_1_2_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_1_1_2_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_3_1_1_2"/>
<font color="Maroon" title="c7">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E2:9_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:9_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A90</font></span>: 
<font color="Maroon" title="c7">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E20">A17</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="c8">i1</font>, <font color="Maroon" title="c9">j1</font>, <font color="Maroon" title="c10">i2</font>, <font color="Maroon" title="c11">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:9_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E27">A91</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:9_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E28">A92</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">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="c1">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="c8">i1</font>,<font color="Maroon" title="c9">j1</font>)
 <span class="kw">and </span><br/><a NAME="E7:9_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E29">A93</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:9_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E30">A94</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c1">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="c10">i2</font>,<font color="Maroon" title="c11">j2</font>)
 <span class="kw">and </span><br/><a NAME="E9:9_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E31">A95</font></span>: 
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">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="c11">j2</font> ) or ( <font color="Maroon" title="c8">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="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="#E3:9_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E10:9_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E32">A96</font></span>: 
 <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a></span>;<br/><a NAME="E11:9_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E33">A97</font></span>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="ref" href="finseq_5.html#T82" title="FINSEQ_5:th.82">FINSEQ_5:82</a></span>;<br/><a NAME="E12:9_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="c7">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="E13:9_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E14:9_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A98</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c7">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="c1">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="c10">i2</font>,<font color="Maroon" title="c11">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E30">A94</font></span></a>, <a class="txt" href="#E11:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E33">A97</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E15:9_1_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E35">A99</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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a></span>;<br/><a NAME="E16:9_1_1_2_1_1_3_1_1_2"/>
<font color="Maroon" title="c7">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E17:9_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E18:9_1_1_2_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A100</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">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="c7">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="c1">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="c8">i1</font>,<font color="Maroon" title="c9">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E11:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E33">A97</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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="9_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="E1:9_1_1_2_1_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">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="c11">j2</font> ) or ( <font color="Maroon" title="c8">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="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="#E9:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E31">A95</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_1_3_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E37">A101</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">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="c11">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E2:9_1_1_2_1_1_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A102</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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E30">A94</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>;<br/><a NAME="E3:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E30">A94</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E37">A101</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="E4:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E10:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E32">A96</font></span></a>, <a class="txt" href="#E15:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E35">A99</font></span></a>, <a class="txt" href="#E18:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E36">A100</font></span></a>, <a class="txt" href="#E14:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E34">A98</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E37">A101</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E38">A102</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_1_1_2_1_1_3_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E37">A103</font></span>: 
( <font color="Maroon" title="c8">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="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E2:9_1_1_2_1_1_3_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A104</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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E30">A94</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><a NAME="E3:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">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="#E3:9_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E30">A94</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E37">A103</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="E4:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E10:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E32">A96</font></span></a>, <a class="txt" href="#E15:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E35">A99</font></span></a>, <a class="txt" href="#E18:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E36">A100</font></span></a>, <a class="txt" href="#E14:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E34">A98</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E37">A103</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E38">A104</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_1_1_2_1_1_3_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_3_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E37">A105</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E2:9_1_1_2_1_1_3_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A106</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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">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="#E3:9_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E30">A94</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><a NAME="E3:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E30">A94</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E37">A105</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="E4:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E10:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E32">A96</font></span></a>, <a class="txt" href="#E15:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E35">A99</font></span></a>, <a class="txt" href="#E18:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E36">A100</font></span></a>, <a class="txt" href="#E14:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E34">A98</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E37">A105</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E38">A106</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="9_1_1_2_1_1_3_1_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2_1_1_3_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E37">A107</font></span>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/><div class="add"><a NAME="E2:9_1_1_2_1_1_3_1_1_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A108</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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c11">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E30">A94</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><a NAME="E3:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E6:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E8:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E30">A94</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E37">A107</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="E4:9_1_1_2_1_1_3_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E5:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E7:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="#E10:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E32">A96</font></span></a>, <a class="txt" href="#E15:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E35">A99</font></span></a>, <a class="txt" href="#E18:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E36">A100</font></span></a>, <a class="txt" href="#E14:9_1_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E34">A98</font></span></a>, <a class="txt" href="#E1:9_1_1_2_1_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E37">A107</font></span></a>, <a class="txt" href="#E2:9_1_1_2_1_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E38">A108</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E20:9_1_1_2_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E8:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E7:9"/><span class="lab"><font color="Green" title="E12">A109</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c6">m</font> &amp; <font color="Maroon" title="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> implies (  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c6">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c6">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">and </span><br/><a NAME="E2:9_2"/><span class="lab"><font color="Green" title="E13">A110</font></span>: 
<font color="Maroon" title="c6">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c6">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c6">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )</span><br/>

<a NAME="E3:9_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">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/>
<span class="kw">hence </span><a NAME="E4:9_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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c6">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c6">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_2"><span class="lab"><font color="Green" title="E13">A110</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:9"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  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="#E7:9"><span class="lab"><font color="Green" title="E12">A109</font></span></a>, <a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E11">A6</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:9"/>
(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> or  not <font color="Maroon" title="c3">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="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> or (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">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="c1">C</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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