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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">C</font> be  <a href="topreal2.html#V1">being_simple_closed_curve</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="E1:15"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">n</font> <a href="jordan1h.html#R1">is_sufficiently_large_for</a> <font color="Maroon">C</font>
 ;<br/>

<b>set </b><font color="Maroon">i1</font> =  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/>
<a NAME="E2:15"/><i><font color="Green">E44</font></i>: 
 <a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E3:15"/><i><font color="Green">E45</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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E4:15"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</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</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/>
<a NAME="E5:15"/><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:15"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1">(<span class="default"><a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 holds <br/> <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Olive">k</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:15_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E2:15_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="jordan11.html#K1" target="_self">ApproxIndex</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan11.html#K2" target="_self">Y-InitStart</a> <font color="Maroon">C</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1"/>
( <span class="p1">(<span class="default"><a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <span class="p1">(<span class="default"><a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1"/><b>then </b><i><font color="Green">E54</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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><a NAME="E3:15_1_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan1h.html#T58">JORDAN1H:58</a></i>;<br/><a NAME="E4:15_1_1_1"/>
<span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E5:15_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</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</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:15_1_1_1"/><i><font color="Green">E57</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp;  <a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</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</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:15_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E8:15_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E9:15_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">k</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</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:15_1_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="jordan1h.html#T58">JORDAN1H:58</a></i>;<br/><a NAME="E11:15_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E12:15_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E13:15_1_1_1"/><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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">k</font></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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <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</font>,<font color="Maroon">n</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">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <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</font>,<font color="Maroon">n</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">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="goboard5.html#T27">GOBOARD5:27</a></i>;<br/><a NAME="E14:15_1_1_1"/><b>then </b><i><font color="Green">E61</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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">k</font> <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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <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="E15:15_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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E16:15_1_1_1"/><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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">k</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><b>set </b><font color="Maroon">W</font> = <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <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> : <font color="Olive">B</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font> </span>}</span> ;<br/><a NAME="E17:15_1_1_1"/><i><font color="Green">E63</font></i>: 
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <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> : <font color="Olive">B</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan2c.html#D5">JORDAN2C:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:15_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <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> : <font color="Olive">B</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font> </span>}</span> 
 <b>and </b><br/><a NAME="E20:15_1_1_1"/><i><font color="Green">E65</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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">k</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T98">ZFMISC_1:98</a></i>;<br/><b>consider </b><font color="Maroon">B</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><b> such that </b><br/><a NAME="E22:15_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E23:15_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">B</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/><a NAME="E24:15_1_1_1"/><i><font color="Green">E84</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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">k</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i>, , <a class="ref" href="jordan1a.html#T46">JORDAN1A:46</a></i>;<br/><a NAME="E25:15_1_1_1"/>
<font color="Maroon">B</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="jordan2c.html#D3">JORDAN2C:def 3</a></i>;<br/><a NAME="E26:15_1_1_1"/><b>then </b><i><font color="Green">E85</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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, , , , <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/><a NAME="E27:15_1_1_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E28:15_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</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2"/><b>then </b>
 <a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="sprect_3.html#T8">SPRECT_3:8</a></i>;<br/><a NAME="E3:15_1_1_2"/><b>then </b>
 <a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_2"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:15"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:15"/>
<span class="p1">(<span class="default"><a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan11.html#K3" target="_self">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/>
<b>hence </b><a NAME="E9:15"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>


</div>
