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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:68"/>
<font color="Maroon">c<sub>1</sub></font> <a href="jordan1h.html#R1" target="_self">is_sufficiently_large_for</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:68"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:68"/><i><font color="Green">E47</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="jordan1h.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:68"/><i><font color="Green">E48</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 
<b>by </b><i><a class="ref" href="jordan1a.html#T49">JORDAN1A:49</a></i>;<br/>
<a NAME="E6:68"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<a NAME="E1:68_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T59" target="_self">Th59</a></i>;<br/>
<b>assume </b><a NAME="E2:68_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan1h.html#E2:68_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1"><b>suppose </b></a><a NAME="E1:68_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:68_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E3:68_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
not  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,0 is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:68_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jordan1a.html#T45">JORDAN1A:45</a></i>;<br/><a NAME="E4:68_1_1_1"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,0 <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:68_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jordan1a.html#T70">JORDAN1A:70</a></i>;<br/><a NAME="E5:68_1_1_1"/><b>then </b>
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E4:68"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan1h.html#E1:68_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan1h.html#E3:68_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T68">XBOOLE_1:68</a></i>;<br/><b>hence </b><a NAME="E6:68_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T28">JORDAN2C:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2"><b>suppose </b></a><a NAME="E1:68_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:68_1_1_2"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/><a NAME="E3:68_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E4:68_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
not  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,1 is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:68_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jordan1a.html#T45">JORDAN1A:45</a></i>;<br/><a NAME="E5:68_1_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,0 <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:68_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jordan1a.html#T70">JORDAN1A:70</a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E6:68_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T73">JORDAN2C:73</a></i>;<br/><a NAME="E7:68_1_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan1h.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E8:68_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E9:68_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1h.html#E7:68_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:68_1_1_2"/><i><font color="Green">E59</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1h.html#E3:68_1_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E11:68_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan1h.html#T59" target="_self">Th59</a></i>;<br/><a NAME="E12:68_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,0</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1h.html#E9:68_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan1h.html#E10:68_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard5.html#T27">GOBOARD5:27</a></i>;<br/><a NAME="E13:68_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,0 <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E14:68_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R2">is_outside_component_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan1h.html#E6:68_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="jordan2c.html#T76">JORDAN2C:76</a></i>;<br/><a NAME="E15:68_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#D4">JORDAN2C:def 4</a></i>;<br/><a NAME="E16:68_1_1_2"/><i><font color="Green">E62</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,1 is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan1h.html#E3:68_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan1h.html#E9:68_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan1a.html#T46">JORDAN1A:46</a></i>;<br/><a NAME="E17:68_1_1_2"/>
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T29">JORDAN2C:29</a></i>;<br/><a NAME="E18:68_1_1_2"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,1 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan1h.html#E4:68"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan1h.html#E1:68_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E19:68_1_1_2"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,1 <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E5:68_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan1h.html#E13:68_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan1h.html#E15:68_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan1h.html#E16:68_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/><a NAME="E20:68_1_1_2"/><b>then </b>
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E4:68"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan1h.html#E1:68_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan1h.html#E4:68_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T68">XBOOLE_1:68</a></i>;<br/><b>hence </b><a NAME="E21:68_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T28">JORDAN2C:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:68"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>proof </b></a><div class="add">

<a NAME="E1:68_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T59" target="_self">Th59</a></i>;<br/>
<b>assume </b><a NAME="E2:68_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:68_2"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan1h.html#E3:68"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan1h.html#E3:68_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1"><b>suppose </b></a><a NAME="E1:68_2_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:68_2_1_1"/><i><font color="Green">E54</font></i>: 
not  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:68_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan1a.html#T45">JORDAN1A:45</a></i>;<br/><a NAME="E3:68_2_1_1"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:68_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan1a.html#T71">JORDAN1A:71</a></i>;<br/><a NAME="E4:68_2_1_1"/><b>then </b>
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E4:68"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan1h.html#E1:68_2_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan1h.html#E2:68_2_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T68">XBOOLE_1:68</a></i>;<br/><b>hence </b><a NAME="E5:68_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T28">JORDAN2C:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_2"><b>suppose </b></a><a NAME="E1:68_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:68_2_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E4:68"><i><font color="Green">E47</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E3:68_2_1_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E4:68_2_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
not  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:68_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan1a.html#T45">JORDAN1A:45</a></i>;<br/><a NAME="E5:68_2_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:68_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan1a.html#T71">JORDAN1A:71</a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E6:68_2_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T73">JORDAN2C:73</a></i>;<br/><a NAME="E7:68_2_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan1h.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E8:68_2_1_2"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/><a NAME="E9:68_2_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E10:68_2_1_2"/>
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E11:68_2_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1h.html#E7:68_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:68_2_1_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><a NAME="E13:68_2_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1h.html#E9:68_2_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E14:68_2_1_2"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1h.html#E9:68_2_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E15:68_2_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan1h.html#T59" target="_self">Th59</a></i>;<br/><a NAME="E16:68_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1h.html#E11:68_2_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan1h.html#E13:68_2_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan1h.html#E14:68_2_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="goboard5.html#T27">GOBOARD5:27</a></i>;<br/><a NAME="E17:68_2_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E18:68_2_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R2">is_outside_component_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan1h.html#E6:68_2_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="jordan2c.html#T76">JORDAN2C:76</a></i>;<br/><a NAME="E19:68_2_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#D4">JORDAN2C:def 4</a></i>;<br/><a NAME="E20:68_2_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan1h.html#E11:68_2_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan1h.html#E13:68_2_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan1a.html#T46">JORDAN1A:46</a></i>;<br/><a NAME="E21:68_2_1_2"/>
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T29">JORDAN2C:29</a></i>;<br/><a NAME="E22:68_2_1_2"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan1h.html#E2:68_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E23:68_2_1_2"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E5:68_2_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan1h.html#E17:68_2_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan1h.html#E19:68_2_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan1h.html#E20:68_2_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/><a NAME="E24:68_2_1_2"/><b>then </b>
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E2:68_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan1h.html#E4:68_2_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T68">XBOOLE_1:68</a></i>;<br/><b>hence </b><a NAME="E25:68_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T28">JORDAN2C:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:68"/><i><font color="Green">E51</font></i>: 
2 <a href="card_4.html#K1">|^</a> 0 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<b>assume </b><a NAME="E9:68"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/>

<a NAME="E10:68"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E11:68"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan1h.html#E3:68"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan1h.html#E5:68"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan1h.html#E8:68"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan1h.html#E10:68"><i><font color="Green">E52</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="jordan1h.html#E11:68"><i><font color="Green">E53</font></i></a>, <a class="ref" href="cqc_the1.html#T5">CQC_THE1:5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_3_1"><b>suppose </b></a><a NAME="E1:68_3_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1h.html#E6:68"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_3_2"><b>suppose </b></a><a NAME="E1:68_3_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><i><font color="Green">E55</font></i>:  <a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,0 = 
<span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default">0 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2

<br/>.= 
1 <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="txt" href="jordan1h.html#E8:68"><i><font color="Green">E51</font></i></a>, <a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,0</span>)</span> <a href="binarith.html#K5">-'</a> 1 = 
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" target="_self">X-SpanStart</a> <font color="Maroon">c<sub>2</sub></font>,0</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
<br/>.= 
2
<b>by </b><i><a class="txt" href="jordan1h.html#E2:68_3_2"><i><font color="Green">E55</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:68_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1h.html#E4:68"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan1h.html#E10:68"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan1h.html#E1:68_3_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jordan1b.html#T19">JORDAN1B:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_3_3"><b>suppose </b></a><a NAME="E1:68_3_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 4 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1h.html#E5:68"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan1h.html#E7:68"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan1h.html#E8:68"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan1h.html#E10:68"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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