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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="connsp_1.html#V2">connected</a> <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>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>given </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:47"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:47"/><i><font color="Green">E33</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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:47"/><i><font color="Green">E34</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>1</sub></font>,<font color="Maroon">c<sub>2</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>2</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="E5:47"/><i><font color="Green">E35</font></i>: 
 <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>1</sub></font>,<font color="Maroon">c<sub>2</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>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E6:47"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1b.html#E2:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1b.html#E3:47"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan1b.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E7:47"/><i><font color="Green">E37</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="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1b.html#E2:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1b.html#E3:47"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan1b.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E8:47"/><i><font color="Green">E38</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="E9:47"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/>

<a NAME="E10:47"/><b>then </b><i><font color="Green">E40</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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 3
 
<b>by </b><i><a class="txt" href="jordan1b.html#E4:47"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan1b.html#E8:47"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E11:47"/><i><font color="Green">E41</font></i>: 
 <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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 3
 
<b>by </b><i><a class="txt" href="jordan1b.html#E5:47"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan1b.html#E8:47"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan1b.html#E9:47"><i><font color="Green">E39</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or <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>4</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 ) or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 4 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="jordan1b.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1b.html#E2:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1b.html#E10:47"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan1b.html#E11:47"><i><font color="Green">E41</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="47_1_1"><b>suppose </b></a><a NAME="E1:47_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1b.html#E2:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1b.html#E3:47"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan1b.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2"><b>suppose </b></a><a NAME="E1:47_1_2"/>
( <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="E2:47_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1b.html#E2:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1b.html#E3:47"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan1b.html#T41" target="_self">Th41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_3"><b>suppose </b></a><a NAME="E1:47_1_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:47_1_3"/><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>1</sub></font>,0</span>)</span>,2,2 <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan1b.html#E3:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan1b.html#E9:47"><i><font color="Green">E39</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>hence </b><a NAME="E3:47_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan1b.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_4"><b>suppose </b></a><a NAME="E1:47_1_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 4 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 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1b.html#E6:47"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan1b.html#E7:47"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan1b.html#E10:47"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan1b.html#E11:47"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
