<?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="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E24</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E25</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:24"/><i><font color="Green">E27</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:24"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</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> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E6:24"/><i><font color="Green">E29</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E4:24"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan9.html#E5:24"><i><font color="Green">E28</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:24"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E9:24"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E10:24"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E11:24"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E12:24"/><i><font color="Green">E34</font></i>: 
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:24"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan9.html#E4:24"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan9.html#E5:24"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E13:24"/><i><font color="Green">E35</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E8:24"><i><font color="Green">E30</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E14:24"/><i><font color="Green">E36</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E10:24"><i><font color="Green">E32</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E12:24"><i><font color="Green">E34</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><div class="add"><a NAME="E2:24_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="ref" href="sppol_1.html#T5">SPPOL_1:5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_1_1"/><b>then </b><i><font color="Green">E39</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="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E3:24_1_1_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan9.html#E9:24"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan9.html#E11:24"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="goboard5.html#T20">GOBOARD5:20</a></i>;<br/><b>hence </b><a NAME="E4:24_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_1_1_1_1"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E2:24_1_1_1_1_2"/>
( 2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binarith.html#K5">-'</a> 1 &amp; 2 <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_1_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E3:24_1_1_1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:24_1_1_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E5:24_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:24_1_1_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan9.html#E9:24"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan9.html#E11:24"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan9.html#E4:24_1_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="goboard5.html#T19">GOBOARD5:19</a></i>;<br/><b>hence </b><a NAME="E7:24_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E3:24_1_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan9.html#E4:24_1_1_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><b>suppose </b></a><a NAME="E1:24_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><div class="add"><a NAME="E2:24_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="ref" href="sppol_1.html#T5">SPPOL_1:5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_2_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:24_1_2_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</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> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E3:24_1_2_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan9.html#E9:24"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan9.html#E11:24"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="goboard5.html#T23">GOBOARD5:23</a></i>;<br/><b>hence </b><a NAME="E4:24_1_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_2_1_1_1"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><a NAME="E2:24_1_2_1_1_2"/>
( 2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binarith.html#K5">-'</a> 1 &amp; 2 <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_2_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E3:24_1_2_1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:24_1_2_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E5:24_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:24_1_2_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:24_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan9.html#E9:24"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan9.html#E11:24"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan9.html#E4:24_1_2_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="goboard5.html#T22">GOBOARD5:22</a></i>;<br/><b>hence </b><a NAME="E7:24_1_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E3:24_1_2_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan9.html#E4:24_1_2_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_3"><b>suppose </b></a><a NAME="E1:24_1_3"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><div class="add"><a NAME="E2:24_1_3"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="sppol_1.html#T5">SPPOL_1:5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_3_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_3_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:24_1_3_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</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> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E3:24_1_3_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan9.html#E9:24"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan9.html#E11:24"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="goboard5.html#T23">GOBOARD5:23</a></i>;<br/><b>hence </b><a NAME="E4:24_1_3_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_3_1_1_1"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_3_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><a NAME="E2:24_1_3_1_1_2"/>
( 2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binarith.html#K5">-'</a> 1 &amp; 2 <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_3_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E3:24_1_3_1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:24_1_3_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E5:24_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:24_1_3_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:24_1_3_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan9.html#E9:24"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan9.html#E11:24"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan9.html#E4:24_1_3_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="goboard5.html#T22">GOBOARD5:22</a></i>;<br/><b>hence </b><a NAME="E7:24_1_3_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E3:24_1_3_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan9.html#E4:24_1_3_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_3"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_4"><b>suppose </b></a><a NAME="E1:24_1_4"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:24_1_4"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="sppol_1.html#T5">SPPOL_1:5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_4_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_4_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:24_1_4_1_1_1"/><b>then </b><i><font color="Green">E39</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="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E3:24_1_4_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan9.html#E9:24"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan9.html#E11:24"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_4"><i><font color="Green">E38</font></i></a>, <a class="ref" href="goboard5.html#T20">GOBOARD5:20</a></i>;<br/><b>hence </b><a NAME="E4:24_1_4_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_4_1_1_1"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_4_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E2:24_1_4_1_1_2"/>
( 2 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binarith.html#K5">-'</a> 1 &amp; 2 <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_4_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E3:24_1_4_1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:24_1_4_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E5:24_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:24_1_4_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan9.html#E9:24"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan9.html#E11:24"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:24_1_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan9.html#E4:24_1_4_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="goboard5.html#T19">GOBOARD5:19</a></i>;<br/><b>hence </b><a NAME="E7:24_1_4_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E13:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E14:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan9.html#E1:24_1_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E3:24_1_4_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan9.html#E4:24_1_4_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_4"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
