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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <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><a NAME="E1:1"/><i><font color="Green">E2</font></i>: 
( 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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="goboard1.html#NM1">Go-board</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <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> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Olive">b<sub>1</sub></font>,1</span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Olive">b<sub>1</sub></font>,1</span>)</span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> ;<br/>
<a NAME="E2:1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:1"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</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><a NAME="E2:1_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E3:1_1"/><i><font color="Green">E7</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> );<br/>
<a NAME="E4:1_1"/><i><font color="Green">E8</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>8</sub></font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>;<br/>

<b>thus </b><a NAME="E1:1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>

<b>thus </b><a NAME="E2:1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>8</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="goboard4.html#E4:1_1"><i><font color="Green">E8</font></i></a>);<br/></i>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E7:1_1"/><i><font color="Green">E10</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a></i>;<br/>
<a NAME="E8:1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a></i>;<br/>
<a NAME="E9:1_1"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="goboard1.html#T22">GOBOARD1:22</a></i>;<br/>
<a NAME="E10:1_1"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="goboard1.html#T23">GOBOARD1:23</a></i>;<br/>
<a NAME="E11:1_1"/>
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E12:1_1"/><b>then </b><i><font color="Green">E14</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> <font color="Maroon">c<sub>5</sub></font> &amp; 0 <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>5</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E13:1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:1_1"/><i><font color="Green">E16</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:1_1"/><i><font color="Green">E17</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E16:1_1"/><i><font color="Green">E18</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E17:1_1"/><i><font color="Green">E19</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E18:1_1"/><i><font color="Green">E20</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1, <font color="Maroon">c<sub>11</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> as   <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>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  );<br/>
<a NAME="E20:1_1"/><i><font color="Green">E21</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:1_1_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E3:1_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<a NAME="E4:1_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E5:1_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E8:1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/>
<a NAME="E9:1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E17:1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>hence </b><a NAME="E10:1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E23:1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>12</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="goboard4.html#E20:1_1"><i><font color="Green">E21</font></i></a>);<br/></i>
<a NAME="E24:1_1"/><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E26:1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>13</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="txt" href="goboard4.html#E21:1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1"><i><font color="Green">E21</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> );<br/>
<a NAME="E27:1_1"/><i><font color="Green">E26</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E28:1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="goboard1.html#T45">GOBOARD1:45</a></i>;<br/>
<a NAME="E29:1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E31:1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>14</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1"><i><font color="Green">E27</font></i></a>);<br/></i>
<a NAME="E32:1_1"/><i><font color="Green">E30</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E33:1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E34:1_1"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_3"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>
<b>hence </b><a NAME="E6:1_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font> as   <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/>
<a NAME="E36:1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> implies <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_4"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_4"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="goboard1.html#T45">GOBOARD1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:1_1_4"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_4"><i><font color="Green">E35</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:1_1_4"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font> as   <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/>
<a NAME="E8:1_1_4"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E9:1_1_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:1_1_4"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="goboard1.html#T49">GOBOARD1:49</a></i>;<br/>
<i><font color="Green">E40</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_4"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:1_1_4"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E11:1_1_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E15:1_1_4"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>20</sub></font> as   <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/>
<a NAME="E17:1_1_4"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E11:1_1_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E18:1_1_4"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_4"><i><font color="Green">E35</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/>
<a NAME="E20:1_1_4"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_4"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E46</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E22:1_1_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E23:1_1_4"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_4"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E24:1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E21:1_1_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E25:1_1_4"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="goboard4.html#E17:1_1_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_4_4"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:1_1_4_4"/><b>then </b>
( <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:1_1_4_4"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E24:1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_4"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E27:1_1_4"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard4.html#E17:1_1_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_4"><i><font color="Green">E44</font></i></a>, <a class="txt" href="goboard4.html#E21:1_1_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E38:1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E39:1_1"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font> as   <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/>
<a NAME="E41:1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1, <font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <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><b> such that </b><br/><a NAME="E44:1_1"/><i><font color="Green">E36</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> holds <br/><font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1">GOBOARD2:sch 1</a>();<br/></i>
<a NAME="E45:1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_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#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1 as   <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>consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_5_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_1"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>22</sub></font> as   <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/><a NAME="E7:1_1_5_1_1"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/><a NAME="E8:1_1_5_1_1"/><b>then </b>
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> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E9:1_1_5_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E10:1_1_5_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:1_1_5_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="goboard1.html#T45">GOBOARD1:45</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:1_1_5_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E15:1_1_5_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E16:1_1_5_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E17:1_1_5_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E18:1_1_5_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E19:1_1_5_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E20:1_1_5_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E17:1_1_5_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_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="E1:1_1_5_1_2_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_5_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> );<br/><a NAME="E2:1_1_5_1_2_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="goboard1.html#T53">GOBOARD1:53</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:1_1_5_1_2_1_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>21</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E43</font></i></a>);<br/></i><b>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font>;<br/><a NAME="E7:1_1_5_1_2_1_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<a NAME="E2:1_1_5_1_2_1_1_2_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="real_1.html#K5">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>6</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] means <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 ;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font> being   <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><b> such that </b><br/><a NAME="E15:1_1_5_1_2_1_1_2"/><i><font color="Green">E47</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="goboard1.html#S1">GOBOARD1:sch 1</a>(<a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2"><i><font color="Green">E46</font></i></a>);<br/></i><a NAME="E16:1_1_5_1_2_1_1_2"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>26</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font> as   <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>consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:1_1_5_1_2_1_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E20:1_1_5_1_2_1_1_2"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>28</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>27</sub></font> as   <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/><a NAME="E22:1_1_5_1_2_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>27</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>29</sub></font> being   <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><b> such that </b><br/><a NAME="E24:1_1_5_1_2_1_1_2"/><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font> holds <br/><font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1">GOBOARD2:sch 1</a>();<br/></i><a NAME="E25:1_1_5_1_2_1_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>30</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font>;<br/><a NAME="E26:1_1_5_1_2_1_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E27:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E28:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E29:1_1_5_1_2_1_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_2"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E58</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_2"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_3"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_4"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_4"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E57</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>31</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>16</sub></font>;<br/><b>thus </b><a NAME="E3:1_1_5_1_2_1_1_2_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</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>5</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_4"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_5"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_5"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>,<font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E58</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>27</sub></font>;<br/><b>thus </b><a NAME="E3:1_1_5_1_2_1_1_2_5"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</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>5</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_5"><i><font color="Green">E61</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_6"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_6"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_6"/><b>then </b>
( <font color="Maroon">c<sub>24</sub></font> is   <a href="nat_1.html#NM1">Nat</a> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>32</sub></font>, <font color="Maroon">c<sub>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:1_1_5_1_2_1_1_2_6"/><i><font color="Green">E63</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</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>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>32</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>33</sub></font>;<br/><b>thus </b><a NAME="E6:1_1_5_1_2_1_1_2_6"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</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>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_6"><i><font color="Green">E62</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_6"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E35:1_1_5_1_2_1_1_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><a NAME="E36:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E33:1_1_5_1_2_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_7"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_7"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_7"/><b>then </b><i><font color="Green">E65</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>16</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</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>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon">c<sub>32</sub></font>, <font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_7"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</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>5</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</sub></font> )
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_7"/><b>then </b>
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_7"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_7"><i><font color="Green">E65</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E38:1_1_5_1_2_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_8"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_8"/><b>then </b><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>,<font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_8"/><b>then </b><i><font color="Green">E67</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>,<font color="Maroon">c<sub>27</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>27</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>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon">c<sub>32</sub></font>, <font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_8"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</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>5</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</sub></font> )
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_8"/><b>then </b>
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_8"><i><font color="Green">E66</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_8"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_9"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_9"/><b>then </b>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_9"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> is   <a href="nat_1.html#NM1">Nat</a> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>32</sub></font>, <font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_9"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</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>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</sub></font> )
 ;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_9"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_9"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font>, <font color="Maroon">c<sub>32</sub></font>, <font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_10"/><i><font color="Green">E67</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>32</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</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>5</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_10"/><b>then </b><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_10"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>16</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_10"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_10"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_10"><i><font color="Green">E68</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_10"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_10"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E7:1_1_5_1_2_1_1_2_10"/><b>then </b>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_10"><i><font color="Green">E67</font></i></a></i>;<br/><b>then </b><i><font color="Green">E72</font></i>: <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_10"><i><font color="Green">E71</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_10"><i><font color="Green">E70</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E9:1_1_5_1_2_1_1_2_10"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>16</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_10"/><b>then </b>
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_10"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_10"><i><font color="Green">E72</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>33</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_10"><i><font color="Green">E69</font></i></a>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E42:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<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>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard4.html#E40:1_1_5_1_2_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><b>set </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>32</sub></font>, <font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_11"/><i><font color="Green">E68</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_11"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_11"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>27</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_11"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_11"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_11"><i><font color="Green">E69</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_11"/><i><font color="Green">E71</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_11"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><i><font color="Green">E73</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_11"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_11"><i><font color="Green">E71</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_11"><i><font color="Green">E71</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_11"><i><font color="Green">E72</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_11"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>27</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_11"/><b>then </b>
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_11"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_11"><i><font color="Green">E73</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_11"><i><font color="Green">E70</font></i></a>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E44:1_1_5_1_2_1_1_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<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>5</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<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>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E39:1_1_5_1_2_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="goboard4.html#E42:1_1_5_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><a NAME="E45:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E36:1_1_5_1_2_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_12"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_12_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_12_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_12_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_12_1_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_12_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_12_1_1"/><i><font color="Green">E72</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_12_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_12_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_12_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_12_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_12_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_12_1_1"><i><font color="Green">E74</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_12_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E8:1_1_5_1_2_1_1_2_12_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_12_1_1"><i><font color="Green">E70</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_12_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_12_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_12_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_12_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_12_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_12_1_2"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_12_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b>
0 <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"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_12_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_12_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E76</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_12_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_12_1_2"><i><font color="Green">E72</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_12_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E15:1_1_5_1_2_1_1_2_12_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_12_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E16:1_1_5_1_2_1_1_2_12_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_12_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E70</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><a NAME="E48:1_1_5_1_2_1_1_2"/>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E49:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_14"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_14_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_14_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_14_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_14_1_1"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_14_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_14_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_14_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_14_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_14_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E38:1_1_5_1_2_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_14_1_1"><i><font color="Green">E75</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_14_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E6:1_1_5_1_2_1_1_2_14_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_14_1_1"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_14_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_14_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_14_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_14_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</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>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_14_1_2"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_14_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_14_1_2"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><i><font color="Green">E79</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_14_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>,<font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_14_1_2"><i><font color="Green">E75</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_14_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E12:1_1_5_1_2_1_1_2_14_1_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_14_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E13:1_1_5_1_2_1_1_2_14_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_14_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E73</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/><a NAME="E52:1_1_5_1_2_1_1_2"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>32</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_5_1_2_1_1_2_16"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:1_1_5_1_2_1_1_2_16"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>33</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font>;<br/>
<b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_16"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>34</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_16"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 
;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_16"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E51:1_1_5_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<b>then </b><i><font color="Green">E79</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_16"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_16"/><i><font color="Green">E80</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_16_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_16_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_16_2_1"/>
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span> or <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_16"><i><font color="Green">E77</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_16"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_16_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_16_2_1_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_5_1_2_1_1_2_16_2_1_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_16_2_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E82</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>34</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E82</font></i></a></i>;<br/><i><font color="Green">E84</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_16"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_16_2_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_16_2_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_16"><i><font color="Green">E75</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_16"><i><font color="Green">E78</font></i></a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_16_2_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E17:1_1_5_1_2_1_1_2_16_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_16"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_16_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_16_2_1_2"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_5_1_2_1_1_2_16_2_1_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_16_2_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>,<font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E82</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>34</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E82</font></i></a></i>;<br/><i><font color="Green">E84</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>27</sub></font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_16"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_16_2_1_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_16_2_1_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_16"><i><font color="Green">E75</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>32</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_16_2_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E17:1_1_5_1_2_1_1_2_16_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_16"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:1_1_5_1_2_1_1_2_16"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E53:1_1_5_1_2_1_1_2"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_17"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E76</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>32</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_17"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_17"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>34</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_17"><i><font color="Green">E77</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>35</sub></font>, <font color="Maroon">c<sub>36</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E81</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>36</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>36</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_17"><i><font color="Green">E80</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>37</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>35</sub></font> as   <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/>
<a NAME="E12:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E82</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>37</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_17"><i><font color="Green">E81</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E15:1_1_5_1_2_1_1_2_17"/><b>then </b>
<font color="Maroon">c<sub>37</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_17"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_17"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E16:1_1_5_1_2_1_1_2_17"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_17"><i><font color="Green">E81</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_17"><i><font color="Green">E83</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E17:1_1_5_1_2_1_1_2_17"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>37</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_17"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_17"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_17"><i><font color="Green">E83</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_17"><i><font color="Green">E84</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E18:1_1_5_1_2_1_1_2_17"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:1_1_5_1_2_1_1_2_17"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_17"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E20:1_1_5_1_2_1_1_2_17"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_17"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_17"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>16</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>5</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_17"/><b>then </b>
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_17"><i><font color="Green">E79</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_17"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_17"><i><font color="Green">E81</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_17"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_17"><i><font color="Green">E79</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_17"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/>
<b>hence </b><a NAME="E24:1_1_5_1_2_1_1_2_17"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E54:1_1_5_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_18"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E76</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>32</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_18"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,<font color="Maroon">c<sub>27</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_18"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>34</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_18"><i><font color="Green">E77</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>35</sub></font>, <font color="Maroon">c<sub>36</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E81</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>36</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>36</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_18"><i><font color="Green">E80</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>37</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>35</sub></font> as   <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/>
<a NAME="E12:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E82</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>37</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_18"><i><font color="Green">E81</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E15:1_1_5_1_2_1_1_2_18"/><b>then </b>
<font color="Maroon">c<sub>37</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_18"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_18"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E16:1_1_5_1_2_1_1_2_18"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_18"><i><font color="Green">E81</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_18"><i><font color="Green">E83</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E17:1_1_5_1_2_1_1_2_18"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>37</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_18"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_18"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_18"><i><font color="Green">E83</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_18"><i><font color="Green">E84</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E18:1_1_5_1_2_1_1_2_18"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:1_1_5_1_2_1_1_2_18"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_18"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E20:1_1_5_1_2_1_1_2_18"/>
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_18"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_18"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,<font color="Maroon">c<sub>27</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>5</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_18"/><b>then </b>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_18"><i><font color="Green">E79</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_18"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_18"><i><font color="Green">E81</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_18"/><b>then </b>
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_18"><i><font color="Green">E85</font></i></a>, <a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/>
<b>hence </b><a NAME="E24:1_1_5_1_2_1_1_2_18"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_18"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><b>then </b><i><font color="Green">E76</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="goboard4.html#E51:1_1_5_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E53:1_1_5_1_2_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>

;<br/><a NAME="E56:1_1_5_1_2_1_1_2"/><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_20"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_5_1_2_1_1_2_20"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_20"/>
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_20"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_20"/><b>then </b>
( <font color="Maroon">c<sub>24</sub></font> is   <a href="nat_1.html#NM1">Nat</a> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:1_1_5_1_2_1_1_2_20"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_20"><i><font color="Green">E78</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E57:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="goboard4.html#E55:1_1_5_1_2_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E58:1_1_5_1_2_1_1_2"/><i><font color="Green">E79</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E59:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="goboard1.html#D10">GOBOARD1:def 10</a></i>;<br/><a NAME="E60:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E47:1_1_5_1_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><i><font color="Green">E82</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E59:1_1_5_1_2_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><b>thus </b><a NAME="E62:1_1_5_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1"/>
( <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E52:1_1_5_1_2_1_1_2"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>7</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> );<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<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"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E85</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
1
;<br/>

<b>thus </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[1]
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="goboard1.html#T22">GOBOARD1:22</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>32</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon">c<sub>32</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(<a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E85</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>32</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><div><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_2"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>32</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>35</sub></font> being   <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><b> such that </b><br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E88</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>35</sub></font> holds <br/><font color="Maroon">c<sub>35</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1">GOBOARD2:sch 1</a>();<br/></i><a NAME="E13:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E89</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E90</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>35</sub></font> holds <br/><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>36</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_3"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>35</sub></font>
 ;<br/>

<a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>32</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>34</sub></font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_3"/>
<font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E92</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>35</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>36</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_4"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>35</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>37</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_4"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>37</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>32</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_1_4"><i><font color="Green">E93</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>37</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>37</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>34</sub></font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>hence </b><a NAME="E8:1_1_5_1_2_1_1_2_22_1_1_4"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_1_4"><i><font color="Green">E93</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>35</sub></font> as   <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>set </b><font color="Maroon">c<sub>37</sub></font> =  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E93</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard1.html#T26">GOBOARD1:26</a></i>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_1"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E94</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="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_1"/>
<font color="Maroon">c<sub>32</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"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E24:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E25:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E89</font></i></a></i>;<br/><a NAME="E26:1_1_5_1_2_1_1_2_22_1_1"/>
(  <a href="matrix_1.html#K9">Col</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; 1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="goboard4.html#E21:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="goboard1.html#T30">GOBOARD1:30</a></i>;<br/><a NAME="E27:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="goboard1.html#T48">GOBOARD1:48</a></i>;<br/><a NAME="E28:1_1_5_1_2_1_1_2_22_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then </b><font color="Maroon">c<sub>36</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E89</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>

;<br/><a NAME="E30:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,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="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="goboard4.html#E21:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="goboard1.html#T30">GOBOARD1:30</a></i>;<br/><a NAME="E31:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E99</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>38</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_6"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E91</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>39</sub></font>, <font color="Maroon">c<sub>40</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_6"/><i><font color="Green">E102</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>39</sub></font>,<font color="Maroon">c<sub>40</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>39</sub></font>,<font color="Maroon">c<sub>40</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon">c<sub>41</sub></font> = <font color="Maroon">c<sub>39</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>42</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>40</sub></font> as   <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/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_6"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E102</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_6"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><b>then </b><i><font color="Green">E104</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>42</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b>
<font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b>
( <font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>40</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>40</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>40</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>40</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E103</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>40</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E102</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E103</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E104</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_6_2"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_6_2"/><b>then </b><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>38</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E103</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_6_2"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_6_2"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_1_6_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_6_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_6_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_6_2"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E13:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b><i><font color="Green">E106</font></i>: 
( 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>40</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>40</sub></font> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b><i><font color="Green">E107</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>40</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E106</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>take </b><font color="Maroon">c<sub>44</sub></font> = <font color="Maroon">c<sub>43</sub></font>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_1_6"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E107</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b><i><font color="Green">E108</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>44</sub></font> &amp; <font color="Maroon">c<sub>44</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="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>40</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b>
<font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E19:1_1_5_1_2_1_1_2_22_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</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> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E61:1_1_5_1_2_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E103</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><a NAME="E20:1_1_5_1_2_1_1_2_22_1_1_6"/>
<font color="Maroon">c<sub>36</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E102</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E103</font></i></a>, <a class="txt" href="goboard4.html#E17:1_1_5_1_2_1_1_2_22_1_1_6"><i><font color="Green">E108</font></i></a>, <a class="ref" href="goboard1.html#T32">GOBOARD1:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>38</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_7"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>36</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b><i><font color="Green">E102</font></i>: 
( <font color="Maroon">c<sub>36</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E88</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>39</sub></font>, <font color="Maroon">c<sub>40</sub></font>, <font color="Maroon">c<sub>41</sub></font>, <font color="Maroon">c<sub>42</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_7"/><i><font color="Green">E103</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>39</sub></font>,<font color="Maroon">c<sub>40</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> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>42</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> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>39</sub></font>,<font color="Maroon">c<sub>40</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>42</sub></font> )
 ;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b><i><font color="Green">E104</font></i>: 
( <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>40</sub></font> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>42</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b><i><font color="Green">E105</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>39</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>41</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>42</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E61:1_1_5_1_2_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_1_7"><i><font color="Green">E102</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_1_7"><i><font color="Green">E103</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_1_7"><i><font color="Green">E104</font></i></a>, <a class="ref" href="goboard1.html#T32">GOBOARD1:32</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b><i><font color="Green">E106</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>39</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>42</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E61:1_1_5_1_2_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_1_7"><i><font color="Green">E104</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_1_7"/>
( <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_1_7"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_1_7"><i><font color="Green">E104</font></i></a></i>;<br/><b>hence </b>1 = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>41</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>42</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_1_7"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_1_7"><i><font color="Green">E106</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>41</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>42</sub></font></span>)</span></span>)</span>

;<br/><br/></div><b>end;</b></div><a NAME="E34:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1
 <b>by </b><i><a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E35:1_1_5_1_2_1_1_2_22_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1_5_1_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard4.html#E50:1_1_5_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="goboard4.html#E58:1_1_5_1_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="goboard4.html#E59:1_1_5_1_2_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E60:1_1_5_1_2_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="goboard1.html#T37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41">GOBOARD1:41</a></i>;<br/><a NAME="E36:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E47:1_1_5_1_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E101</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>38</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>36</sub></font></span>)</span>;<br/><a NAME="E38:1_1_5_1_2_1_1_2_22_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E39:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E36:1_1_5_1_2_1_1_2_22_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E40:1_1_5_1_2_1_1_2_22_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E57:1_1_5_1_2_1_1_2"><i><font color="Green">E78</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>32</sub></font> =  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2"/><i><font color="Green">E84</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1"/>
( <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard4.html#E52:1_1_5_1_2_1_1_2"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>33</sub></font> being   <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><b> such that </b><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,1</span>)</span> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="goboard1.html#T51">GOBOARD1:51</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><i><font color="Green">E87</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard1.html#T26">GOBOARD1:26</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1_5_1_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="goboard4.html#E50:1_1_5_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="goboard4.html#E58:1_1_5_1_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="goboard4.html#E59:1_1_5_1_2_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E60:1_1_5_1_2_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard1.html#T37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41">GOBOARD1:41</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E47:1_1_5_1_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_1"><i><font color="Green">E87</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>34</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E57:1_1_5_1_2_1_1_2"><i><font color="Green">E78</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="goboard4.html#E40:1_1_5_1_2_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>33</sub></font> being   <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><b> such that </b><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E87</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard1.html#T52">GOBOARD1:52</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>34</sub></font> being   <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><b> such that </b><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,1</span>)</span> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="goboard1.html#T51">GOBOARD1:51</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>7</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> );<br/><div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">c<sub>35</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"/>
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>36</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>take </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>36</sub></font>;<br/><b>thus </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon">c<sub>37</sub></font>]
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>8</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> );<br/><div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">c<sub>35</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"/>
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>36</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>take </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>36</sub></font>;<br/><b>thus </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon">c<sub>37</sub></font>]
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon">c<sub>35</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E89</font></i></a>);<br/></i><b>consider </b><font color="Maroon">c<sub>36</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon">c<sub>36</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E90</font></i></a>);<br/></i><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E94</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1"/>
( <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>36</sub></font> or <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>36</sub></font> )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E93</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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1"/>
<font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1"/>
1 <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"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_2"/>
<font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_2"/>
1 <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"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>37</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/>
<font color="Maroon">c<sub>37</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>38</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>, <font color="Maroon">c<sub>39</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>40</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font> as   <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/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E97</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><b>then </b><i><font color="Green">E98</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>38</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><b>consider </b><font color="Maroon">c<sub>41</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>39</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E100</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>39</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>39</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>42</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>41</sub></font> as   <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/><a NAME="E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E101</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E29:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>39</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">c<sub>43</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>38</sub></font> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/>
<font color="Maroon">c<sub>38</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><b>then </b><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>43</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E105</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E107</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>43</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E105</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>41</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E104</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E107</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>44</sub></font>, <font color="Maroon">c<sub>45</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E109</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>44</sub></font>,<font color="Maroon">c<sub>45</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>44</sub></font>,<font color="Maroon">c<sub>45</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>44</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E107</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E108</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E109</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E109</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1"/>
( (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>44</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> ) or (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E110</font></i></a>, <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/><i><font color="Green">E112</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>44</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><i><font color="Green">E113</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>38</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>38</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>38</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>44</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>44</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E109</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="goboard4.html#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><i><font color="Green">E112</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>39</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>39</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E109</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>39</sub></font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><i><font color="Green">E104</font></i></a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>43</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>43</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>44</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>43</sub></font> as   <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/><a NAME="E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>43</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">c<sub>45</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>38</sub></font> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><i><font color="Green">E112</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>38</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>38</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/>
<font color="Maroon">c<sub>38</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><b>then </b><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>45</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E112</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E114</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>45</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E112</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>43</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><b>then </b><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E110</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E114</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>46</sub></font>, <font color="Maroon">c<sub>47</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E116</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>46</sub></font>,<font color="Maroon">c<sub>47</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>46</sub></font>,<font color="Maroon">c<sub>47</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>46</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E114</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E115</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E116</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E116</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1"/>
( (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>46</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> ) or (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E117</font></i></a>, <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/><i><font color="Green">E119</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>46</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><i><font color="Green">E120</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>38</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>38</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>38</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E118</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E116</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E118</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"><i><font color="Green">E119</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E118</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="goboard4.html#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><i><font color="Green">E119</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E118</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>40</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>47</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E116</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E118</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>26</sub></font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>41</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>45</sub></font> being   <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><b> such that </b><br/><a NAME="E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E111</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font> holds <br/><font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1">GOBOARD2:sch 1</a>();<br/></i><a NAME="E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E112</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>43</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>46</sub></font> being   <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><b> such that </b><br/><a NAME="E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E113</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font> holds <br/><font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1">GOBOARD2:sch 1</a>();<br/></i><a NAME="E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E114</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><i><font color="Green">E115</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>47</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><b>then </b><i><font color="Green">E116</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>47</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><b>then </b>
<font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><b>then </b>
<font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><i><font color="Green">E116</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E116</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>47</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>47</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/>
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E117</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>47</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/><b>then </b><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>47</sub></font>,<font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E48:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E115</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>48</sub></font> = <font color="Maroon">c<sub>47</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>49</sub></font> = <font color="Maroon">c<sub>41</sub></font>;<br/><b>thus </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>48</sub></font>,<font color="Maroon">c<sub>49</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>5</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>48</sub></font>,<font color="Maroon">c<sub>49</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"><i><font color="Green">E118</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E118</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>47</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"/><b>then </b><i><font color="Green">E119</font></i>: 
( <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span>,<font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E116</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>48</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>49</sub></font> = <font color="Maroon">c<sub>43</sub></font>;<br/><b>thus </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>48</sub></font>,<font color="Maroon">c<sub>49</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>5</sub></font> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>48</sub></font>,<font color="Maroon">c<sub>49</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"><i><font color="Green">E119</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>47</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/>
( <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><b>then </b><i><font color="Green">E120</font></i>: 
( <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>47</sub></font>,<font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E48:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E115</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><b>then </b><i><font color="Green">E121</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>47</sub></font>,<font color="Maroon">c<sub>41</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>41</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>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon">c<sub>48</sub></font>, <font color="Maroon">c<sub>49</sub></font>, <font color="Maroon">c<sub>50</sub></font>, <font color="Maroon">c<sub>51</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>48</sub></font>,<font color="Maroon">c<sub>49</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>50</sub></font>,<font color="Maroon">c<sub>51</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>5</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>48</sub></font>,<font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>50</sub></font>,<font color="Maroon">c<sub>51</sub></font> )
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><b>then </b>
( <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><i><font color="Green">E120</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><i><font color="Green">E121</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>48</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>51</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><i><font color="Green">E120</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>47</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"/>
( <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"/><b>then </b><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span>,<font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E116</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"/><b>then </b><i><font color="Green">E122</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span>,<font color="Maroon">c<sub>43</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>43</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>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon">c<sub>48</sub></font>, <font color="Maroon">c<sub>49</sub></font>, <font color="Maroon">c<sub>50</sub></font>, <font color="Maroon">c<sub>51</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>48</sub></font>,<font color="Maroon">c<sub>49</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>50</sub></font>,<font color="Maroon">c<sub>51</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>5</sub></font> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>48</sub></font>,<font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>50</sub></font>,<font color="Maroon">c<sub>51</sub></font> )
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"/><b>then </b>
( <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"><i><font color="Green">E121</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"><i><font color="Green">E122</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>48</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>51</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>9</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> );<br/><a NAME="E54:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[0]
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E55:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E122</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>47</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon">c<sub>47</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>48</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>48</sub></font></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1"/>
( <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"><i><font color="Green">E127</font></i></a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>39</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"><i><font color="Green">E126</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><i><font color="Green">E125</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>47</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>47</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>25</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><i><font color="Green">E124</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>47</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>49</sub></font>, <font color="Maroon">c<sub>50</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E128</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>49</sub></font>,<font color="Maroon">c<sub>50</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>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>49</sub></font>,<font color="Maroon">c<sub>50</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green">E61</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>51</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>49</sub></font> as   <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/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E129</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>51</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>51</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E131</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>51</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>51</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>50</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>51</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>51</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><i><font color="Green">E123</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>52</sub></font>, <font color="Maroon">c<sub>53</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E135</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>52</sub></font>,<font color="Maroon">c<sub>53</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>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>52</sub></font>,<font color="Maroon">c<sub>53</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><i><font color="Green">E124</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>54</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font> as   <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/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E136</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>54</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>54</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E138</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>54</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>54</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>53</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E136</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E137</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>54</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>53</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E137</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>54</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E136</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E137</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E138</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>49</sub></font> )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E134</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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E127</font></i></a></i>;<br/><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E127</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E127</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>51</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"><i><font color="Green">E140</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"><i><font color="Green">E141</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"><i><font color="Green">E142</font></i></a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><i><font color="Green">E124</font></i></a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>12</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>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"><i><font color="Green">E144</font></i></a>, <a class="ref" href="goboard1.html#T44">GOBOARD1:44</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><i><font color="Green">E125</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"><i><font color="Green">E145</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>49</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>49</sub></font></span>)</span> or for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><i><font color="Green">E124</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E133</font></i></a>, <a class="ref" href="goboard1.html#T43">GOBOARD1:43</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>49</sub></font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><i><font color="Green">E125</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E141</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>52</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E137</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E139</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>52</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2"><i><font color="Green">E141</font></i></a>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>52</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">c<sub>49</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><i><font color="Green">E125</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E137</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>52</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><i><font color="Green">E125</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E137</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E123</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="goboard4.html#E54:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="goboard4.html#E55:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E122</font></i></a>);<br/></i><a NAME="E57:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E125</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>47</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E88</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>48</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>49</sub></font>, <font color="Maroon">c<sub>50</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E128</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>49</sub></font>,<font color="Maroon">c<sub>50</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>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>49</sub></font>,<font color="Maroon">c<sub>50</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E127</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>51</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>49</sub></font> as   <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/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E129</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>51</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>51</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E128</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E131</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b>
<font color="Maroon">c<sub>51</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>51</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>50</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E130</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>51</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E130</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>51</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E131</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E127</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E130</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>52</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E126</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E134</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>52</sub></font>,<font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E48:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E133</font></i></a></i>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>52</sub></font>,<font color="Maroon">c<sub>41</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>5</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b><i><font color="Green">E136</font></i>: 
( <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E127</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E135</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>52</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/>
<b>hence </b><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><i><font color="Green">E136</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>10</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> );<br/><a NAME="E58:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">S<sub>10</sub></font>[0]
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E59:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E126</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>10</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>10</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>47</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">S<sub>10</sub></font>[<font color="Maroon">c<sub>47</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>48</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>48</sub></font></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1"/>
( <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>39</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"><i><font color="Green">E130</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"/>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"><i><font color="Green">E129</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>47</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>47</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>25</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"><i><font color="Green">E128</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E131</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>47</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>49</sub></font>, <font color="Maroon">c<sub>50</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E133</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>49</sub></font>,<font color="Maroon">c<sub>50</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>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>49</sub></font>,<font color="Maroon">c<sub>50</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green">E61</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>51</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>49</sub></font> as   <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/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E134</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>51</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>51</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E133</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E136</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>51</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>51</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>50</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>51</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>51</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E136</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"><i><font color="Green">E127</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>52</sub></font>, <font color="Maroon">c<sub>53</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E139</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>52</sub></font>,<font color="Maroon">c<sub>53</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>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>52</sub></font>,<font color="Maroon">c<sub>53</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"><i><font color="Green">E128</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>54</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font> as   <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/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E140</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>54</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>54</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E142</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>54</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>54</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>53</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E141</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>54</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>53</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E141</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>54</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E141</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E142</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>48</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E141</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> or <font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font> )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E138</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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> is   <a href="nat_1.html#NM1">Nat</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E132</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E137</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"><i><font color="Green">E145</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"><i><font color="Green">E146</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"><i><font color="Green">E147</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E137</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"><i><font color="Green">E145</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"><i><font color="Green">E146</font></i></a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E38:1_1_5_1_2_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>49</sub></font></span>)</span> or for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E137</font></i></a>, <a class="ref" href="goboard1.html#T43">GOBOARD1:43</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>49</sub></font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>52</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E141</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E143</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>52</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>52</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E144</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><i><font color="Green">E144</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"/>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E60:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>10</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="goboard4.html#E58:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="goboard4.html#E59:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E126</font></i></a>);<br/></i><a NAME="E61:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E129</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>47</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E88</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>48</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>49</sub></font>, <font color="Maroon">c<sub>50</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E132</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>49</sub></font>,<font color="Maroon">c<sub>50</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>5</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>49</sub></font>,<font color="Maroon">c<sub>50</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E131</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>51</sub></font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>49</sub></font> as   <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/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E133</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>51</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>51</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E132</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E135</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
<font color="Maroon">c<sub>51</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>51</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>50</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>51</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E134</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>51</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E135</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E60:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E134</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>52</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E130</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font></span>)</span>,<font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E137</font></i></a></i>;<br/>
<a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font></span>)</span>,<font color="Maroon">c<sub>43</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>5</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b><i><font color="Green">E139</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E138</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>52</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E137</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>52</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E136</font></i></a>, <a class="txt" href="goboard4.html#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><i><font color="Green">E139</font></i></a>, <a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1"/>
( <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>36</sub></font> or <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>35</sub></font> )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E93</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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>47</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
( <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E129</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b>
<font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>48</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>35</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>49</sub></font> = <font color="Maroon">c<sub>35</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>50</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/>

<a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>50</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>50</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>50</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>36</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"><i><font color="Green">E132</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>11</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] means <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E131</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>11</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 ;<br/><b>consider </b><font color="Maroon">c<sub>50</sub></font> being   <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><b> such that </b><br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E132</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>48</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> holds <br/><font color="Maroon">S<sub>11</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="goboard1.html#S1">GOBOARD1:sch 1</a>(<a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E131</font></i></a>);<br/></i><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E133</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>51</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
( <font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E129</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b>
( <font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>48</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>52</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
( <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E133</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
( <font color="Maroon">c<sub>49</sub></font> is   <a href="nat_1.html#NM1">Nat</a> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E130</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>53</sub></font> = <font color="Maroon">c<sub>49</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><b>consider </b><font color="Maroon">c<sub>54</sub></font>, <font color="Maroon">c<sub>55</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/><i><font color="Green">E137</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>54</sub></font>,<font color="Maroon">c<sub>55</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>54</sub></font>,<font color="Maroon">c<sub>55</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"><i><font color="Green">E136</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon">c<sub>56</sub></font> = <font color="Maroon">c<sub>54</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>57</sub></font> = <font color="Maroon">c<sub>55</sub></font>;<br/><b>thus </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>56</sub></font>,<font color="Maroon">c<sub>57</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>5</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>56</sub></font>,<font color="Maroon">c<sub>57</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"><i><font color="Green">E137</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E50:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/><div><i><font color="Green">E137</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>52</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/>
( <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> &amp; <font color="Maroon">c<sub>52</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; <font color="Maroon">c<sub>52</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>52</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>52</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>52</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E133</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><b>then </b><i><font color="Green">E138</font></i>: 
( <font color="Maroon">c<sub>49</sub></font> is   <a href="nat_1.html#NM1">Nat</a> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>52</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>52</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E130</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>53</sub></font>, <font color="Maroon">c<sub>54</sub></font>, <font color="Maroon">c<sub>55</sub></font>, <font color="Maroon">c<sub>56</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>53</sub></font>,<font color="Maroon">c<sub>54</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>55</sub></font>,<font color="Maroon">c<sub>56</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>5</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>53</sub></font>,<font color="Maroon">c<sub>54</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>52</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>55</sub></font>,<font color="Maroon">c<sub>56</sub></font> )
 ;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>55</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>54</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>56</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><i><font color="Green">E138</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E138</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>52</sub></font>, <font color="Maroon">c<sub>53</sub></font>, <font color="Maroon">c<sub>54</sub></font>, <font color="Maroon">c<sub>55</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/><i><font color="Green">E139</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>52</sub></font>,<font color="Maroon">c<sub>53</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>54</sub></font>,<font color="Maroon">c<sub>55</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>5</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>52</sub></font>,<font color="Maroon">c<sub>53</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>54</sub></font>,<font color="Maroon">c<sub>55</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/><b>then </b><i><font color="Green">E140</font></i>: 
( <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E48:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E115</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>41</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"><i><font color="Green">E139</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"><i><font color="Green">E140</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"><i><font color="Green">E139</font></i></a></i>;<br/><b>then </b><i><font color="Green">E142</font></i>: <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E130</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>41</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>41</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/><b>then </b>
( <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"><i><font color="Green">E139</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"><i><font color="Green">E142</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>52</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>54</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>55</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"><i><font color="Green">E141</font></i></a>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>52</sub></font> = <font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font>;<br/><div><i><font color="Green">E139</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>53</sub></font>, <font color="Maroon">c<sub>54</sub></font>, <font color="Maroon">c<sub>55</sub></font>, <font color="Maroon">c<sub>56</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><i><font color="Green">E140</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>53</sub></font>,<font color="Maroon">c<sub>54</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>55</sub></font>,<font color="Maroon">c<sub>56</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>53</sub></font>,<font color="Maroon">c<sub>54</sub></font> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>55</sub></font>,<font color="Maroon">c<sub>56</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E116</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>43</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><b>then </b><i><font color="Green">E142</font></i>: 
( <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><i><font color="Green">E140</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><i><font color="Green">E141</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><i><font color="Green">E143</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><i><font color="Green">E144</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><i><font color="Green">E143</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>48</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><i><font color="Green">E143</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>43</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><i><font color="Green">E143</font></i></a></i>
;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>43</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><b>then </b>
( <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><i><font color="Green">E140</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><i><font color="Green">E144</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>55</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>54</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>56</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><i><font color="Green">E142</font></i></a>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E140</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E51:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E135</font></i></a>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a><br/>

<a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<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>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="goboard4.html#E52:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="txt" href="goboard4.html#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E137</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E138</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/>
<b>hence </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<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>5</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<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>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E53:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E139</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E141</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/><b>then </b>
<font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/><b>then </b><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>50</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/><i><font color="Green">E144</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E129</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"><i><font color="Green">E144</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"><i><font color="Green">E144</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"><i><font color="Green">E145</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"><i><font color="Green">E142</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"><i><font color="Green">E142</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b><i><font color="Green">E144</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><i><font color="Green">E145</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b>
0 <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"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>41</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"><i><font color="Green">E144</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>41</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"><i><font color="Green">E147</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E142</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E143</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E134</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><i><font color="Green">E144</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"/><b>then </b>
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"/><b>then </b><i><font color="Green">E147</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>36</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"><i><font color="Green">E146</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"><i><font color="Green">E146</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"><i><font color="Green">E147</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E136</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"><i><font color="Green">E147</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"><i><font color="Green">E147</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"><i><font color="Green">E145</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</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>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><i><font color="Green">E150</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"><i><font color="Green">E150</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>,<font color="Maroon">c<sub>43</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"><i><font color="Green">E147</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>43</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"><i><font color="Green">E149</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E145</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/><a NAME="E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E146</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>53</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/><i><font color="Green">E148</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>54</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font>;<br/>
<b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>55</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E145</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<b>then </b><i><font color="Green">E150</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/><i><font color="Green">E151</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>41</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>43</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1"/>
( <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span> or <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><i><font color="Green">E149</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><i><font color="Green">E150</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/>
<font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
( <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>56</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>56</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>56</sub></font>,<font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E153</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>57</sub></font> = <font color="Maroon">c<sub>55</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E153</font></i></a></i>;<br/><i><font color="Green">E155</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>42</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><i><font color="Green">E151</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E48:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E155</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/>
<font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>56</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E156</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>57</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>56</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E154</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E155</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E156</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>57</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E156</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b><i><font color="Green">E158</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>53</sub></font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><i><font color="Green">E147</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E152</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E157</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E155</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><i><font color="Green">E158</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/>
<font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
( <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font> &amp; <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>56</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font> &amp; <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>56</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>56</sub></font></span>)</span>,<font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>57</sub></font> = <font color="Maroon">c<sub>55</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><i><font color="Green">E155</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>44</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><i><font color="Green">E151</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>44</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E155</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/>
<font color="Maroon">c<sub>44</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>56</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>56</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>57</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>56</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E154</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E155</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>57</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>44</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>53</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><i><font color="Green">E147</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E152</font></i></a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>48</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>48</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>48</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>53</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E155</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E158</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><i><font color="Green">E159</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/>
contradiction
 ;<br/>


</div><b>end;</b></div><i><font color="Green">E147</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="goboard4.html#E61:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E145</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E57:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font></span>)</span>

;<br/><a NAME="E29:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E148</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E56:1_1_5_1_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>53</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/>
<font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>54</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> &amp; <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>54</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/>
( <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>54</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"><i><font color="Green">E149</font></i></a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/><b>then </b>
( <font color="Maroon">c<sub>49</sub></font> is   <a href="nat_1.html#NM1">Nat</a> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>54</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E130</font></i></a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/><b>then </b>
<font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"><i><font color="Green">E149</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/>
<font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E147</font></i></a>, <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E150</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E143</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><i><font color="Green">E151</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>53</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>50</sub></font> &amp; <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"><i><font color="Green">E151</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>54</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>53</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><i><font color="Green">E154</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>50</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><b>then </b><i><font color="Green">E155</font></i>: 
( <font color="Maroon">c<sub>50</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>54</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>49</sub></font> is   <a href="nat_1.html#NM1">Nat</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>54</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>54</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"><i><font color="Green">E153</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>55</sub></font> = <font color="Maroon">c<sub>49</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>55</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>54</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font> &amp; <font color="Maroon">c<sub>55</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"><i><font color="Green">E152</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"><i><font color="Green">E153</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"><i><font color="Green">E155</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><b>then </b><i><font color="Green">E156</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>55</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>54</sub></font> &amp; <font color="Maroon">c<sub>54</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"><i><font color="Green">E153</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"><i><font color="Green">E154</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><b>then </b>
<font color="Maroon">c<sub>55</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>54</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>55</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>48</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goboard4.html#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"><i><font color="Green">E156</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E146</font></i></a></i>;<br/><a NAME="E36:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E151</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard4.html#E58:1_1_5_1_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="goboard4.html#E59:1_1_5_1_2_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E140</font></i></a>, <a class="txt" href="goboard4.html#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E141</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E144</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E150</font></i></a>, <a class="ref" href="goboard1.html#T37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41">GOBOARD1:41</a></i>;<br/><a NAME="E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard1.html#T26">GOBOARD1:26</a></i>;<br/><a NAME="E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E142</font></i></a>, <a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E151</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>53</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>;<br/><a NAME="E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
<font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>50</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E152</font></i></a></i>;<br/><b>hence </b><a NAME="E42:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><i><font color="Green">E149</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>35</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>47</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
( <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
<font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>48</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>36</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>49</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/>
<font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/>

<a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><b>then </b><i><font color="Green">E131</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>49</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E131</font></i>: 
for <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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>49</sub></font>, <font color="Maroon">c<sub>50</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font>
 <b>and </b><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>49</sub></font>
 ;<br/>

<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><i><font color="Green">E134</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>51</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>49</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><i><font color="Green">E135</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>49</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>49</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>49</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>35</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"><i><font color="Green">E133</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>11</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E132</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>11</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>49</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_3"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>50</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>49</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E130</font></i></a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font>
;<br/>

<b>thus </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_3"/>
<font color="Maroon">S<sub>11</sub></font>[<font color="Maroon">c<sub>49</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>50</sub></font>]
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_3"><i><font color="Green">E133</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>49</sub></font> being   <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><b> such that </b><br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E133</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>48</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> holds <br/><font color="Maroon">S<sub>11</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="goboard1.html#S1">GOBOARD1:sch 1</a>(<a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E132</font></i></a>);<br/></i><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E134</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>50</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
( <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
( <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>48</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>51</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>52</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>51</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"><i><font color="Green">E136</font></i></a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>51</sub></font> is   <a href="nat_1.html#NM1">Nat</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E131</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>53</sub></font>, <font color="Maroon">c<sub>54</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><i><font color="Green">E138</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>53</sub></font>,<font color="Maroon">c<sub>54</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>53</sub></font>,<font color="Maroon">c<sub>54</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon">c<sub>55</sub></font> = <font color="Maroon">c<sub>53</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>56</sub></font> = <font color="Maroon">c<sub>54</sub></font>;<br/><b>thus </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>55</sub></font>,<font color="Maroon">c<sub>56</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>5</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>55</sub></font>,<font color="Maroon">c<sub>56</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"><i><font color="Green">E137</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"><i><font color="Green">E138</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E50:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E51:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green">E137</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>51</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>51</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>52</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>51</sub></font>, <font color="Maroon">c<sub>53</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/>
( <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; <font color="Maroon">c<sub>51</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>53</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>51</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><i><font color="Green">E138</font></i></a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><b>then </b><i><font color="Green">E139</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>52</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>53</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E131</font></i></a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">c<sub>53</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>51</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>54</sub></font>, <font color="Maroon">c<sub>55</sub></font>, <font color="Maroon">c<sub>56</sub></font>, <font color="Maroon">c<sub>57</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>54</sub></font>,<font color="Maroon">c<sub>55</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>56</sub></font>,<font color="Maroon">c<sub>57</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>5</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>54</sub></font>,<font color="Maroon">c<sub>55</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>56</sub></font>,<font color="Maroon">c<sub>57</sub></font> )
 ;<br/><b>hence </b>1 = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>56</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>54</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>57</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>55</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><i><font color="Green">E139</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><i><font color="Green">E140</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>54</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>56</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>55</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>57</sub></font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>54</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>56</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>55</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>57</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>54</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>56</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>55</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>57</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>51</sub></font>, <font color="Maroon">c<sub>52</sub></font>, <font color="Maroon">c<sub>53</sub></font>, <font color="Maroon">c<sub>54</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><i><font color="Green">E139</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>51</sub></font>,<font color="Maroon">c<sub>52</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>53</sub></font>,<font color="Maroon">c<sub>54</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>5</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>51</sub></font>,<font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>53</sub></font>,<font color="Maroon">c<sub>54</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><b>then </b><i><font color="Green">E140</font></i>: 
( <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E48:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E115</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>41</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><i><font color="Green">E139</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><i><font color="Green">E140</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>55</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><i><font color="Green">E142</font></i>: <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>55</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><i><font color="Green">E139</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>55</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><i><font color="Green">E139</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>41</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>41</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><b>then </b>
( <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><i><font color="Green">E139</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><i><font color="Green">E142</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>53</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>52</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>54</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><i><font color="Green">E141</font></i></a>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E139</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<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>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E52:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="txt" href="goboard4.html#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E137</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><b>set </b><font color="Maroon">c<sub>51</sub></font> = <font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>52</sub></font>, <font color="Maroon">c<sub>53</sub></font>, <font color="Maroon">c<sub>54</sub></font>, <font color="Maroon">c<sub>55</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><i><font color="Green">E140</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>52</sub></font>,<font color="Maroon">c<sub>53</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>54</sub></font>,<font color="Maroon">c<sub>55</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>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>52</sub></font>,<font color="Maroon">c<sub>53</sub></font> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>54</sub></font>,<font color="Maroon">c<sub>55</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E116</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>43</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><b>then </b><i><font color="Green">E142</font></i>: 
( <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"><i><font color="Green">E140</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"><i><font color="Green">E141</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><i><font color="Green">E143</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>56</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>48</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><i><font color="Green">E144</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"><i><font color="Green">E143</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>56</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"><i><font color="Green">E143</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>56</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"><i><font color="Green">E143</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>43</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="goboard4.html#E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>43</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><b>then </b>
( <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"><i><font color="Green">E140</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"><i><font color="Green">E144</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>52</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>54</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>55</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"><i><font color="Green">E142</font></i></a>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<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>5</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<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>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E53:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E140</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E136</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><div><i><font color="Green">E141</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/><b>then </b>
<font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/><b>then </b><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/><i><font color="Green">E144</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"><i><font color="Green">E144</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E138</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"><i><font color="Green">E144</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"><i><font color="Green">E145</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"><i><font color="Green">E142</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"><i><font color="Green">E142</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b><i><font color="Green">E144</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><i><font color="Green">E145</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b>
0 <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"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>41</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"><i><font color="Green">E144</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>41</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"><i><font color="Green">E147</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E142</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><a NAME="E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><i><font color="Green">E144</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/><b>then </b>
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/><i><font color="Green">E148</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"><i><font color="Green">E146</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"><i><font color="Green">E146</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"><i><font color="Green">E148</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"><i><font color="Green">E147</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"><i><font color="Green">E148</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>36</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"><i><font color="Green">E148</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"><i><font color="Green">E145</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</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>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</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>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><i><font color="Green">E150</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"><i><font color="Green">E150</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>,<font color="Maroon">c<sub>43</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"><i><font color="Green">E147</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>43</sub></font>
<b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"><i><font color="Green">E149</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E145</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/><a NAME="E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E146</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>52</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><i><font color="Green">E148</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>53</sub></font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font>;<br/>
<b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>54</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<b>then </b><i><font color="Green">E150</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><i><font color="Green">E151</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>41</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>43</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1"/>
( <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span></span>)</span> or <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><i><font color="Green">E149</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><i><font color="Green">E150</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/>
<font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
( <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>55</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>55</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>55</sub></font>,<font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E153</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>56</sub></font> = <font color="Maroon">c<sub>54</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><i><font color="Green">E155</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font>
 ;<br/><i><font color="Green">E156</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>42</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><i><font color="Green">E151</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E48:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E156</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/>
<font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>55</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E157</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>55</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E154</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E156</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E157</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E157</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b><i><font color="Green">E159</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E135</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><i><font color="Green">E147</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E152</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E155</font></i></a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E158</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E156</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><i><font color="Green">E159</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/>
<font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
( <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>46</sub></font> &amp; <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>55</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>46</sub></font> &amp; <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>46</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>55</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font></span>)</span>,<font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>56</sub></font> = <font color="Maroon">c<sub>54</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>48</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>57</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>48</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><i><font color="Green">E156</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>44</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><i><font color="Green">E151</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>44</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/>
<font color="Maroon">c<sub>44</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>55</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E154</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E156</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>44</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><i><font color="Green">E147</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E152</font></i></a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/>
( <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>57</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>57</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>57</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E155</font></i></a></i>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="goboard4.html#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="goboard4.html#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E158</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>52</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E155</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E156</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><i><font color="Green">E159</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/>
contradiction
 ;<br/>


</div><b>end;</b></div><i><font color="Green">E147</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="goboard4.html#E61:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E57:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font></span>)</span>

;<br/><a NAME="E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E148</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E56:1_1_5_1_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>52</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/>
<font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>53</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>53</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>54</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>53</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"><i><font color="Green">E149</font></i></a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/>
<font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>54</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"><i><font color="Green">E149</font></i></a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>54</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"><i><font color="Green">E149</font></i></a></i>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/><b>then </b>
<font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"><i><font color="Green">E149</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>hence </b><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/>
<font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E147</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E36:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E150</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E142</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E143</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><i><font color="Green">E151</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>52</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><i><font color="Green">E151</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>53</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>54</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>53</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><i><font color="Green">E153</font></i></a></i>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>54</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>54</sub></font> &amp; <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><i><font color="Green">E153</font></i></a></i>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>53</sub></font> &amp; <font color="Maroon">c<sub>53</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>48</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><i><font color="Green">E153</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><b>then </b><i><font color="Green">E155</font></i>: 
( <font color="Maroon">c<sub>54</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>54</sub></font> )
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><i><font color="Green">E154</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/>
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>38</sub></font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><i><font color="Green">E152</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><i><font color="Green">E153</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><i><font color="Green">E154</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>54</sub></font> &amp; <font color="Maroon">c<sub>54</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E104</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><i><font color="Green">E155</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><i><font color="Green">E156</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E146</font></i></a></i>;<br/><a NAME="E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E40:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E151</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard4.html#E58:1_1_5_1_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="goboard4.html#E59:1_1_5_1_2_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E140</font></i></a>, <a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E141</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E144</font></i></a>, <a class="txt" href="goboard4.html#E36:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E150</font></i></a>, <a class="ref" href="goboard1.html#T37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41">GOBOARD1:41</a></i>;<br/><a NAME="E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard1.html#T26">GOBOARD1:26</a></i>;<br/><a NAME="E42:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E142</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E40:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E151</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>52</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>;<br/><a NAME="E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
<font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>46</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E42:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E152</font></i></a></i>;<br/><b>hence </b><a NAME="E45:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><i><font color="Green">E149</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="E63:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_5_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E47:1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:1"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="goboard4.html#E3:1"><i><font color="Green">E4</font></i></a>);<br/></i>
<div><i><font color="Green">E6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</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><a NAME="E1:1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> )
 ;<br/><a NAME="E2:1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1"><i><font color="Green">E5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:1"/>
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font> <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="E7:1"/><b>then </b>
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<b>hence </b><a NAME="E8:1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="goboard4.html#E5:1"><i><font color="Green">E6</font></i></a></i>;<br/>


</div>
