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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>assume </b><a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f2</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">G</font>,1</span>)</span> &amp; <font color="Maroon" title="c2">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">G</font>,1</span>)</span> &amp; <font color="Maroon" title="c3">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">G1</font> being  <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a><br/> for <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">G1</font> &amp; $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">g2</font> &amp; <font color="Olive" title="b2">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Olive" title="b3">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Olive" title="b2">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Olive" title="b1">G1</font>,1</span>)</span> &amp; <font color="Olive" title="b2">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Olive" title="b1">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G1</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b3">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Olive" title="b1">G1</font>,1</span>)</span> &amp; <font color="Olive" title="b3">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Olive" title="b1">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">G1</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<a NAME="E2:1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:1"/><i><font color="Green" title="E4">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c5">G1</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/><b>let </b><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>assume </b><a NAME="E2:1_1"/><i><font color="Green" title="E6">A5</font></i>: 
( <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E3:1_1"/><i><font color="Green" title="E7">A6</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> );<br/>
<a NAME="E4:1_1"/><i><font color="Green" title="E8">A7</font></i>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</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" title="c8">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font>;<br/>

<b>thus </b><a NAME="E1:1_1_1"/>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>

<b>thus </b><a NAME="E2:1_1_1"/>
<font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:1_1"/><i><font color="Green" title="E9">A8</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">m</font>] &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="goboard4.html#E4:1_1"><i><font color="Green" title="E8">A7</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font>;<br/>
<a NAME="E7:1_1"/><i><font color="Green" title="E10">A9</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="goboard1.html#T38" title="GOBOARD1:th.38">GOBOARD1:38</a></i>;<br/>
<a NAME="E8:1_1"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="goboard1.html#T38" title="GOBOARD1:th.38">GOBOARD1:38</a></i>;<br/>
<a NAME="E9:1_1"/><i><font color="Green" title="E12">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="goboard1.html#T22" title="GOBOARD1:th.22">GOBOARD1:22</a></i>;<br/>
<a NAME="E10:1_1"/><i><font color="Green" title="E13">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="goboard1.html#T23" title="GOBOARD1:th.23">GOBOARD1:23</a></i>;<br/>
<a NAME="E11:1_1"/>
( 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp; 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 
<b>by </b><i><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E12:1_1"/><b>then </b><i><font color="Green" title="E14">A13</font></i>: 
( 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp; 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E13:1_1"/><b>then </b><i><font color="Green" title="E15">A14</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:1_1"/><i><font color="Green" title="E16">A15</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:1_1"/><i><font color="Green" title="E17">A16</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E16:1_1"/><i><font color="Green" title="E18">A17</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E17:1_1"/><i><font color="Green" title="E19">A18</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E18:1_1"/><i><font color="Green" title="E20">A19</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">L1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1, <font color="Maroon" title="c11">Ll</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,$1</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  );<br/>
<a NAME="E20:1_1"/><i><font color="Green" title="E21">A20</font></i>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</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" title="c12">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:1_1_2"/><i><font color="Green" title="E22">A21</font></i>: 
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E3:1_1_2"/><i><font color="Green" title="E23">A22</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">i</font>
;<br/>

<a NAME="E4:1_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E5:1_1_2"/><b>then </b><i><font color="Green" title="E24">A23</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:1_1_2"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_2"><i><font color="Green" title="E23">A22</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_2"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></i>;<br/>
<a NAME="E9:1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="goboard4.html#E17:1_1"><i><font color="Green" title="E19">A18</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_2"><i><font color="Green" title="E24">A23</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:1_1"/><i><font color="Green" title="E22">A24</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">mi</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E23:1_1"/><i><font color="Green" title="E23">A25</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">mi</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="goboard4.html#E20:1_1"><i><font color="Green" title="E21">A20</font></i></a>);<br/></i>
<a NAME="E24:1_1"/><i><font color="Green" title="E24">A26</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">ma</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E26:1_1"/><i><font color="Green" title="E25">A27</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">ma</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="goboard4.html#E21:1_1"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1"><i><font color="Green" title="E21">A20</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font></span>)</span> );<br/>
<a NAME="E27:1_1"/><i><font color="Green" title="E26">A28</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font> &amp; <font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E28:1_1"/><b>then </b><i><font color="Green" title="E27">A29</font></i>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="goboard1.html#T45" title="GOBOARD1:th.45">GOBOARD1:45</a></i>;<br/>
<a NAME="E29:1_1"/><i><font color="Green" title="E28">A30</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">ma1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E31:1_1"/><i><font color="Green" title="E29">A31</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c14">ma1</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">ma1</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1"><i><font color="Green" title="E27">A29</font></i></a>);<br/></i>
<a NAME="E32:1_1"/><i><font color="Green" title="E30">A32</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E33:1_1"/><b>then </b><i><font color="Green" title="E31">A33</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E34:1_1"/><i><font color="Green" title="E32">A34</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</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" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_3"/><i><font color="Green" title="E33">A35</font></i>: 
( <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:1_1_3"/>
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green" title="E31">A33</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_3"><i><font color="Green" title="E33">A35</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:1_1_3"/><b>then </b>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g2</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_3"><i><font color="Green" title="E33">A35</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/>
<b>hence </b><a NAME="E6:1_1_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c15">Lmi</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E36:1_1"/><i><font color="Green" title="E33">A36</font></i>: 
( <font color="Maroon" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> implies <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E34">A37</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c16">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_4"/><i><font color="Green" title="E35">A38</font></i>: 
( <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green" title="E26">A28</font></i></a>, <a class="ref" href="goboard1.html#T45" title="GOBOARD1:th.45">GOBOARD1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:1_1_4"/><i><font color="Green" title="E36">A39</font></i>: 
( <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font></span>)</span> &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Lmi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">i</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_4"><i><font color="Green" title="E35">A38</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:1_1_4"/><i><font color="Green" title="E37">A40</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">Ci</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c17">i</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E8:1_1_4"/><i><font color="Green" title="E38">A41</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c17">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E9:1_1_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_4"><i><font color="Green" title="E36">A39</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:1_1_4"/><i><font color="Green" title="E39">A42</font></i>: 
( <font color="Maroon" title="c19">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c17">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="goboard1.html#T49" title="GOBOARD1:th.49">GOBOARD1:49</a></i>;<br/>
<i><font color="Green" title="E40">A43</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c17">i</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E8:1_1_4"><i><font color="Green" title="E38">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c20">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E14:1_1_4"/><i><font color="Green" title="E41">A44</font></i>: 
( <font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">Ci</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Ci</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c20">j</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E11:1_1_4"><i><font color="Green" title="E39">A42</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E15:1_1_4"/><i><font color="Green" title="E42">A45</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">Ci</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">Ci</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">Lj</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c20">j</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E17:1_1_4"/><i><font color="Green" title="E43">A46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E11:1_1_4"><i><font color="Green" title="E39">A42</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E18:1_1_4"/><i><font color="Green" title="E44">A47</font></i>: 
<font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_4"><i><font color="Green" title="E35">A38</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">Lj</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
;<br/>
<a NAME="E20:1_1_4"/><b>then </b><i><font color="Green" title="E45">A48</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">Lj</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_4"><i><font color="Green" title="E37">A40</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green" title="E46">A49</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">m</font> = 
<font color="Maroon" title="c18">Ci</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font>
<b>by </b><i><a class="txt" href="goboard4.html#E14:1_1_4"><i><font color="Green" title="E41">A44</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c21">Lj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
<b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_4"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_4"><i><font color="Green" title="E37">A40</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_4"><i><font color="Green" title="E38">A41</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_4"><i><font color="Green" title="E40">A43</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_4"><i><font color="Green" title="E41">A44</font></i></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></i>
<br/>.= 
<font color="Maroon" title="c21">Lj</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c17">i</font>
<b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_4"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_4"><i><font color="Green" title="E45">A48</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E22:1_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">Lj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E23:1_1_4"/><b>then </b>
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">Lj</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E5:1_1_4"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_4"><i><font color="Green" title="E37">A40</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_4"><i><font color="Green" title="E38">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">Lj</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E21:1_1_4"><i><font color="Green" title="E46">A49</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E25:1_1_4"/><b>then </b><i><font color="Green" title="E47">A50</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c20">j</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="goboard4.html#E17:1_1_4"><i><font color="Green" title="E43">A46</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c20">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">mi</font>
 ;<br/><a NAME="E2:1_1_4_4"/><b>then </b>
( <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">mi</font> or <font color="Maroon" title="c20">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">mi</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E24:1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_4"><i><font color="Green" title="E34">A37</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_4"><i><font color="Green" title="E38">A41</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_4"><i><font color="Green" title="E41">A44</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_4"><i><font color="Green" title="E42">A45</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_4"><i><font color="Green" title="E47">A50</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_4"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="goboard4.html#E17:1_1_4"><i><font color="Green" title="E43">A46</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_4"><i><font color="Green" title="E44">A47</font></i></a>, <a class="txt" href="goboard4.html#E21:1_1_4"><i><font color="Green" title="E46">A49</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c16">k1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E38:1_1"/><i><font color="Green" title="E34">A51</font></i>: 
( <font color="Maroon" title="c16">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">Lmi</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">ma1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Lmi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">k1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E39:1_1"/><i><font color="Green" title="E35">A52</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">Lmi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">Lmi</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">Ck1</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c16">k1</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E41:1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">l1</font> = <font color="Maroon" title="c12">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1, <font color="Maroon" title="c19">l2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">ma</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> $1,<font color="Maroon" title="c16">k1</font>;<br/>
<b>consider </b><font color="Maroon" title="c20">h1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E44:1_1"/><i><font color="Green" title="E36">A53</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">l1</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h1</font> holds <br/><font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1" title="GOBOARD2:sch.1">GOBOARD2:sch 1</a>();<br/></i>
<a NAME="E45:1_1"/><i><font color="Green" title="E37">A54</font></i>: 
<font color="Maroon" title="c16">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green" title="E34">A51</font></i></a>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green" title="E35">A52</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="E38">A55</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c21">c1</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1 as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c22">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:1_1_5_1_1"/><i><font color="Green" title="E39">A56</font></i>: 
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">c1</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">x</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_1"/><i><font color="Green" title="E40">A57</font></i>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">c1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">c1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">lx</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c22">x</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E7:1_1_5_1_1"/>
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></i>;<br/><a NAME="E8:1_1_5_1_1"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E9:1_1_5_1_1"/><b>then </b><i><font color="Green" title="E41">A58</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">lx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">lx</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">lx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_1"><i><font color="Green" title="E39">A56</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_1"><i><font color="Green" title="E40">A57</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E10:1_1_5_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">x</font> &amp; <font color="Maroon" title="c22">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_1"><i><font color="Green" title="E39">A56</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_1"><i><font color="Green" title="E40">A57</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:1_1_5_1_1"/><i><font color="Green" title="E42">A59</font></i>: 
( <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">lx</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="goboard1.html#T45" title="GOBOARD1:th.45">GOBOARD1:45</a></i>;<br/><b>consider </b><font color="Maroon" title="c25">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E14:1_1_5_1_1"/><i><font color="Green" title="E43">A60</font></i>: 
( <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">lx</font> &amp; <font color="Maroon" title="c23">lx</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c24">m</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_1"><i><font color="Green" title="E42">A59</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E15:1_1_5_1_1"/><i><font color="Green" title="E44">A61</font></i>: 
( <font color="Maroon" title="c23">lx</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">lx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">y</font> &amp; <font color="Maroon" title="c21">c1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_1"><i><font color="Green" title="E39">A56</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_1"><i><font color="Green" title="E43">A60</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E16:1_1_5_1_1"/><i><font color="Green" title="E45">A62</font></i>: 
( <font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_1"><i><font color="Green" title="E38">A55</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_1"><i><font color="Green" title="E41">A58</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_1"><i><font color="Green" title="E43">A60</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E17:1_1_5_1_1"/><b>then </b><i><font color="Green" title="E46">A63</font></i>: 
<font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_1"><i><font color="Green" title="E39">A56</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1_5_1_1"><i><font color="Green" title="E41">A58</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_1"><i><font color="Green" title="E43">A60</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_1"><i><font color="Green" title="E44">A61</font></i></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></i>;<br/><a NAME="E18:1_1_5_1_1"/><i><font color="Green" title="E47">A64</font></i>: 
<font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_1"><i><font color="Green" title="E42">A59</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E19:1_1_5_1_1"/>
<font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_1"><i><font color="Green" title="E45">A62</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E17:1_1_5_1_1"><i><font color="Green" title="E46">A63</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1_5_1_1"><i><font color="Green" title="E47">A64</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="E38">A65</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2"/><b>then </b><i><font color="Green" title="E39">A66</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</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" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> or <font color="Maroon" title="c12">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">ma</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" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green" title="E33">A36</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" title="E40">A67</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">ma</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c14">ma1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c13">ma</font></span>)</span> );<br/><a NAME="E2:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E41">A68</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E42">A69</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">ma</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green" title="E40">A67</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="E43">A70</font></i>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="ref" href="goboard1.html#T53" title="GOBOARD1:th.53">GOBOARD1:53</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">mi1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E44">A71</font></i>: 
( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c21">mi1</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c21">mi1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green" title="E43">A70</font></i></a>);<br/></i><b>set </b><font color="Maroon" title="c22">f1</font> = <font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font>;<br/><a NAME="E7:1_1_5_1_2_1_1_2"/>
( <font color="Maroon" title="c14">ma1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">mi1</font> &amp; <font color="Maroon" title="c21">mi1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">mi1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E44">A71</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">ma1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">mi1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c23">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c21">mi1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c14">ma1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">ma1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">w</font> = <font color="Maroon" title="c14">ma1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E45">A72</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</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" title="c25">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_1"/><i><font color="Green" title="E46">A73</font></i>: 
<font color="Maroon" title="c25">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font>
 ;<br/>

<a NAME="E2:1_1_5_1_2_1_1_2_1"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">ma1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">ma1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n</font> &amp; <font color="Maroon" title="c25">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">l</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" title="E46">A73</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="E47">A74</font></i>: 
( 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">ma1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c25">n</font> &amp; <font color="Maroon" title="c25">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">ma1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">ma1</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="c25">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">ma1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">mi1</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c25">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c21">mi1</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" title="E47">A74</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c25">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c25">n</font></span>)</span> &amp; <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c25">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green" title="E44">A71</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> $1</span>)</span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E46">A75</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">p</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c25">f</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E47">A76</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">l</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font>] ) )
 <b>from </b><i><a class="ref" href="goboard1.html#S1" title="GOBOARD1:sch.1">GOBOARD1:sch 1</a>(<a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2"><i><font color="Green" title="E46">A75</font></i></a>);<br/></i><a NAME="E16:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E48">A77</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">Lma</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c13">ma</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c27">k2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E19:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E49">A78</font></i>: 
( <font color="Maroon" title="c27">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">Lma</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c21">mi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">k2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green" title="E44">A71</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E20:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E50">A79</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">Lma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">Lma</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">Ck2</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c27">k2</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E22:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E51">A80</font></i>: 
<font color="Maroon" title="c27">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E49">A78</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green" title="E50">A79</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> $1</span>)</span>,<font color="Maroon" title="c27">k2</font>;<br/><b>consider </b><font color="Maroon" title="c29">h2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E52">A81</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l2</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font> holds <br/><font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1" title="GOBOARD2:sch.1">GOBOARD2:sch 1</a>();<br/></i><a NAME="E25:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E53">A82</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c30">ff</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font>;<br/><a NAME="E26:1_1_5_1_2_1_1_2"/>
( <font color="Maroon" title="c14">ma1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">mi1</font> &amp; <font color="Maroon" title="c21">mi1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">mi1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E44">A71</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E27:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E54">A83</font></i>: 
( <font color="Maroon" title="c14">ma1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">mi1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E55">A84</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">l</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">l</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E29:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E56">A85</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></i>;<br/><div><i><font color="Green" title="E57">A86</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" title="c31">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_2"/>
<font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h1</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_2"/><b>then </b><i><font color="Green" title="E58">A87</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font> &amp; <font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">l1</font> &amp; <font color="Maroon" title="c18">l1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font> &amp; <font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green" title="E36">A53</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E58">A87</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_2"/>
<font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</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" title="E58">A87</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E58">A88</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" title="c31">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_3"/>
<font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font> &amp; <font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l2</font> &amp; <font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c13">ma</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green" title="E52">A81</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c13">ma</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">l2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_3"/>
<font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E59">A89</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" title="c31">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_4"/>
<font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h1</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_4"/><b>then </b><i><font color="Green" title="E60">A90</font></i>: 
( <font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c31">n</font>,<font color="Maroon" title="c16">k1</font> &amp; <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green" title="E36">A53</font></i></a>, <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green" title="E57">A86</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c32">i</font> = <font color="Maroon" title="c31">n</font>;<br/><b>take </b><font color="Maroon" title="c33">k1</font> = <font color="Maroon" title="c16">k1</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">i</font>,<font color="Maroon" title="c33">k1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">i</font>,<font color="Maroon" title="c33">k1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green" title="E37">A54</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_4"><i><font color="Green" title="E60">A90</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E60">A91</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" title="c31">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_5"/>
<font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_5"/><b>then </b><i><font color="Green" title="E61">A92</font></i>: 
( <font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span>,<font color="Maroon" title="c27">k2</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green" title="E52">A81</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green" title="E58">A88</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c32">m</font> = <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font>;<br/><b>take </b><font color="Maroon" title="c33">k2</font> = <font color="Maroon" title="c27">k2</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">m</font>,<font color="Maroon" title="c33">k2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">m</font>,<font color="Maroon" title="c33">k2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green" title="E51">A80</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_5"><i><font color="Green" title="E61">A92</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E61">A93</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" title="c31">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_6"/>
<font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_6"/><b>then </b><i><font color="Green" title="E62">A94</font></i>: 
( <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</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" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_6"/><b>then </b>
( <font color="Maroon" title="c24">w</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</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" title="E45">A72</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c32">i</font>, <font color="Maroon" title="c33">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:1_1_5_1_2_1_1_2_6"/><i><font color="Green" title="E63">A95</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">i</font>,<font color="Maroon" title="c33">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">i</font>,<font color="Maroon" title="c33">j</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon" title="c34">i</font> = <font color="Maroon" title="c32">i</font>;<br/><b>take </b><font color="Maroon" title="c35">j</font> = <font color="Maroon" title="c33">j</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">i</font>,<font color="Maroon" title="c35">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c34">i</font>,<font color="Maroon" title="c35">j</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green" title="E45">A72</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_6"><i><font color="Green" title="E62">A94</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_6"><i><font color="Green" title="E63">A95</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" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2"><i><font color="Green" title="E59">A89</font></i></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></i>;<br/><a NAME="E36:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E62">A96</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E33:1_1_5_1_2_1_1_2"><i><font color="Green" title="E60">A91</font></i></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green" title="E63">A97</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" title="c31">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_7"/>
( <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h1</font> &amp; <font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h1</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_7"/><b>then </b><i><font color="Green" title="E64">A98</font></i>: 
( <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c31">n</font>,<font color="Maroon" title="c16">k1</font> &amp; <font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c16">k1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green" title="E36">A53</font></i></a>, <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green" title="E57">A86</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_7"/><b>then </b><i><font color="Green" title="E65">A99</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">n</font>,<font color="Maroon" title="c16">k1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c16">k1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green" title="E37">A54</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon" title="c32">i1</font>, <font color="Maroon" title="c33">i2</font>, <font color="Maroon" title="c34">j1</font>, <font color="Maroon" title="c35">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</font> &amp; <font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</font> )
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_7"/><b>then </b>
( <font color="Maroon" title="c32">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">n</font> &amp; <font color="Maroon" title="c33">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k1</font> &amp; <font color="Maroon" title="c34">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c35">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k1</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E64">A98</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_7"><i><font color="Green" title="E65">A99</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c34">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c35">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">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" title="E64">A100</font></i>: 
<font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">mi1</font>
 ;<br/><div><i><font color="Green" title="E65">A101</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" title="c31">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_8"/>
( <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font> &amp; <font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_8"/><b>then </b><i><font color="Green" title="E66">A102</font></i>: 
( <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span>,<font color="Maroon" title="c27">k2</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c27">k2</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E52">A81</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green" title="E58">A88</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_8"/><b>then </b><i><font color="Green" title="E67">A103</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span>,<font color="Maroon" title="c27">k2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c27">k2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green" title="E51">A80</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon" title="c32">i1</font>, <font color="Maroon" title="c33">i2</font>, <font color="Maroon" title="c34">j1</font>, <font color="Maroon" title="c35">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</font> )
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_8"/><b>then </b>
( <font color="Maroon" title="c32">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font> &amp; <font color="Maroon" title="c33">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">k2</font> &amp; <font color="Maroon" title="c34">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c35">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">k2</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E66">A102</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_8"><i><font color="Green" title="E67">A103</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c34">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c35">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><i><font color="Green" title="E66">A104</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" title="c31">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_9"/>
( <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> &amp; <font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_9"/><b>then </b>
( <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> &amp; <font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_9"/><b>then </b><i><font color="Green" title="E67">A105</font></i>: 
( <font color="Maroon" title="c24">w</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E45">A72</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c32">i1</font>, <font color="Maroon" title="c33">i2</font>, <font color="Maroon" title="c34">j1</font>, <font color="Maroon" title="c35">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</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" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c34">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c35">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_9"><i><font color="Green" title="E67">A105</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">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" title="c31">i1</font>, <font color="Maroon" title="c32">i2</font>, <font color="Maroon" title="c33">j1</font>, <font color="Maroon" title="c34">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_10"/><i><font color="Green" title="E67">A106</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c31">i1</font>,<font color="Maroon" title="c32">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c33">j1</font>,<font color="Maroon" title="c34">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c31">i1</font>,<font color="Maroon" title="c32">i2</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c33">j1</font>,<font color="Maroon" title="c34">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h1</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_10"/><b>then </b><i><font color="Green" title="E68">A107</font></i>: 
( <font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">l1</font>,<font color="Maroon" title="c16">k1</font> &amp; <font color="Maroon" title="c18">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green" title="E36">A53</font></i></a>, <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green" title="E57">A86</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">l1</font>,<font color="Maroon" title="c16">k1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green" title="E37">A54</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E69">A108</font></i>: 
( <font color="Maroon" title="c31">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">l1</font> &amp; <font color="Maroon" title="c32">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k1</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E67">A106</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_10"><i><font color="Green" title="E68">A107</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_10"/><i><font color="Green" title="E70">A109</font></i>: 
<font color="Maroon" title="c15">Lmi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Lmi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green" title="E34">A51</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_10"/><i><font color="Green" title="E71">A110</font></i>: 
<font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">ma1</font>
 ;<br/><a NAME="E7:1_1_5_1_2_1_1_2_10"/><b>then </b>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">ma1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_10"><i><font color="Green" title="E67">A106</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E72">A111</font></i>: <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">ma1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green" title="E45">A72</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_10"><i><font color="Green" title="E71">A110</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">mi</font>,<font color="Maroon" title="c16">k1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green" title="E34">A51</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green" title="E37">A54</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_10"><i><font color="Green" title="E70">A109</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">mi</font>,<font color="Maroon" title="c16">k1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green" title="E37">A54</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_10"/><b>then </b>
( <font color="Maroon" title="c33">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</font> &amp; <font color="Maroon" title="c34">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k1</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" title="E67">A106</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_10"><i><font color="Green" title="E72">A111</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c33">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c34">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">mi</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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" title="E69">A108</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">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" title="E67">A112</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> holds <br/>for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b4">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b5">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E63">A97</font></i></a>, <a class="txt" href="goboard4.html#E40:1_1_5_1_2_1_1_2"><i><font color="Green" title="E66">A104</font></i></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></i>;<br/><b>set </b><font color="Maroon" title="c31">hf</font> = <font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</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" title="c32">i1</font>, <font color="Maroon" title="c33">i2</font>, <font color="Maroon" title="c34">j1</font>, <font color="Maroon" title="c35">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_11"/><i><font color="Green" title="E68">A113</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_11"/><b>then </b><i><font color="Green" title="E69">A114</font></i>: 
( <font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c27">k2</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green" title="E52">A81</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green" title="E58">A88</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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c27">k2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green" title="E51">A80</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E70">A115</font></i>: 
( <font color="Maroon" title="c34">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c35">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">k2</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" title="E68">A113</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2_11"><i><font color="Green" title="E69">A114</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_11"/><i><font color="Green" title="E71">A116</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E48">A77</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green" title="E55">A84</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_11"/><i><font color="Green" title="E72">A117</font></i>: 
<font color="Maroon" title="c26">Lma</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green" title="E49">A78</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><i><font color="Green" title="E73">A118</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</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" title="E71">A116</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">l</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" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_11"><i><font color="Green" title="E71">A116</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c21">mi1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green" title="E45">A72</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_11"><i><font color="Green" title="E71">A116</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">ma</font>,<font color="Maroon" title="c27">k2</font>
<b>by </b><i><a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green" title="E49">A78</font></i></a>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green" title="E51">A80</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2_11"><i><font color="Green" title="E72">A117</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">ma</font>,<font color="Maroon" title="c27">k2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green" title="E51">A80</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_11"/><b>then </b>
( <font color="Maroon" title="c32">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> &amp; <font color="Maroon" title="c33">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">k2</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" title="E68">A113</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_11"><i><font color="Green" title="E73">A118</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c34">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c35">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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" title="E70">A115</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c13">ma</font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">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" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E65">A101</font></i></a>, <a class="txt" href="goboard4.html#E42:1_1_5_1_2_1_1_2"><i><font color="Green" title="E67">A112</font></i></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></i>;<br/><a NAME="E45:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E68">A119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E36:1_1_5_1_2_1_1_2"><i><font color="Green" title="E62">A96</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><div><i><font color="Green" title="E69">A120</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" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="E70">A121</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c20">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green" title="E36">A53</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">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" title="E71">A122</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_12_1_1"/><i><font color="Green" title="E72">A123</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">ma1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green" title="E55">A84</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E73">A124</font></i>: 
<font color="Maroon" title="c14">ma1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c21">mi1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green" title="E44">A71</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_12_1_1"/><i><font color="Green" title="E74">A125</font></i>: 
<font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">ma1</font>
 ;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="E48">A77</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" title="E71">A122</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" title="E72">A123</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">ma1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</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" title="E72">A123</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" title="E74">A125</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">ma1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green" title="E44">A71</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" title="E73">A124</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">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" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green" title="E29">A31</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" title="E70">A121</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" title="E70">A126</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">mi</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" title="E70">A126</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b><i><font color="Green" title="E71">A127</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">l1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="E72">A128</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green" title="E36">A53</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_12_1_2"/><i><font color="Green" title="E73">A129</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_12_1_2"/><i><font color="Green" title="E74">A130</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E75">A131</font></i>: 
<font color="Maroon" title="c10">L1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green" title="E37">A54</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</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" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green" title="E36">A53</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" title="E71">A127</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" title="E73">A129</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E76">A132</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="E72">A128</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c16">k1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green" title="E36">A53</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" title="E72">A128</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">L1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">k1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green" title="E37">A54</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" title="E75">A131</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E15:1_1_5_1_2_1_1_2_12_1_2"/>
<font color="Maroon" title="c16">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">L1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green" title="E37">A54</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" title="E74">A130</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</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" title="E76">A132</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green" title="E70">A133</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
;<br/><a NAME="E48:1_1_5_1_2_1_1_2"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E49:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E71">A134</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</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" title="E55">A84</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><div><i><font color="Green" title="E72">A135</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" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> or <font color="Maroon" title="c13">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E73">A136</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="c29">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green" title="E52">A81</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">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" title="E74">A137</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">mi1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green" title="E44">A71</font></i></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="E75">A138</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> &amp; <font color="Maroon" title="c21">mi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c21">mi1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green" title="E54">A83</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green" title="E55">A84</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</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" title="E74">A137</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c23">l</font>
<b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</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" title="E74">A137</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" title="E75">A138</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c21">mi1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</font></i></a>, <a class="txt" href="goboard4.html#E38:1_1_5_1_2_1_1_2"><i><font color="Green" title="E64">A100</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" title="E75">A138</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c21">mi1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green" title="E44">A71</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" title="E75">A138</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">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" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E44">A71</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" title="E73">A136</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" title="E73">A139</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_14_1_2"/>
<font color="Maroon" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c13">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E73">A139</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b><i><font color="Green" title="E74">A140</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="E75">A141</font></i>: 
<font color="Maroon" title="c19">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_14_1_2"/><i><font color="Green" title="E76">A142</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green" title="E52">A81</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" title="E74">A140</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_14_1_2"/><i><font color="Green" title="E77">A143</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c27">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Ll</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green" title="E51">A80</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E78">A144</font></i>: 
<font color="Maroon" title="c11">Ll</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k2</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><i><font color="Green" title="E79">A145</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">l2</font>
<b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green" title="E52">A81</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" title="E76">A142</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">l2</font></span>)</span>,<font color="Maroon" title="c27">k2</font>
<b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green" title="E52">A81</font></i></a>, <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green" title="E53">A82</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" title="E75">A141</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">Ll</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">k2</font>
<b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green" title="E51">A80</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" title="E78">A144</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E12:1_1_5_1_2_1_1_2_14_1_2"/>
<font color="Maroon" title="c27">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Ll</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green" title="E51">A80</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" title="E77">A143</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E79">A145</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green" title="E73">A146</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
;<br/><a NAME="E52:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E74">A147</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="c32">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E75">A148</font></i>: 
<font color="Maroon" title="c32">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <b>and </b><br/><a NAME="E2:1_1_5_1_2_1_1_2_16"/><i><font color="Green" title="E76">A149</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c33">gg</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font>;<br/>
<b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_16"/><i><font color="Green" title="E77">A150</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c34">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font></span>)</span></span>)</span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_16"/><i><font color="Green" title="E78">A151</font></i>: 
<font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">ma1</font>
 
;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_16"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</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" title="E73">A146</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/>
<b>then </b><i><font color="Green" title="E79">A152</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</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" title="E76">A149</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_16"/><i><font color="Green" title="E80">A153</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c16">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c27">k2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font></span>)</span></span>)</span> or <font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</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" title="E77">A150</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1_5_1_2_1_1_2_16"><i><font color="Green" title="E79">A152</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</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" title="E81">A154</font></i>: 
( <font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font> &amp; <font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c35">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E82">A155</font></i>: 
( <font color="Maroon" title="c35">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h1</font> &amp; <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_16_2_1_1"/><i><font color="Green" title="E83">A156</font></i>: 
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c35">i</font>,<font color="Maroon" title="c16">k1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green" title="E36">A53</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" title="E82">A155</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c36">y</font> = <font color="Maroon" title="c34">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E82">A155</font></i></a></i>;<br/><i><font color="Green" title="E84">A157</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c16">k1</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E80">A153</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E85">A158</font></i>: 
<font color="Maroon" title="c35">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">Ck1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green" title="E57">A86</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" title="E82">A155</font></i></a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_16_2_1_1"/><i><font color="Green" title="E86">A159</font></i>: 
( <font color="Maroon" title="c15">Lmi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Lmi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k1</font> &amp; <font color="Maroon" title="c17">Ck1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Ck1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">mi</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green" title="E34">A51</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" title="E84">A157</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c17">Ck1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Ck1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</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" title="E85">A158</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c36">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Ck1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">i</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" title="E83">A156</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" title="E84">A157</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" title="E85">A158</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="c36">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">Ck1</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" title="E85">A158</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E87">A160</font></i>: 
( <font color="Maroon" title="c16">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">k</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green" title="E37">A54</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green" title="E55">A84</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_16"><i><font color="Green" title="E75">A148</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" title="E81">A154</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">ma1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">ma1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E45">A72</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_16"><i><font color="Green" title="E78">A151</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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Ck1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green" title="E34">A51</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green" title="E37">A54</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" title="E86">A159</font></i></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font></span>)</span> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</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" title="E84">A157</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" title="E87">A160</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E76">A149</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</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" title="E81">A161</font></i>: 
( <font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font> &amp; <font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c35">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E82">A162</font></i>: 
( <font color="Maroon" title="c35">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font> &amp; <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_16_2_1_2"/><i><font color="Green" title="E83">A163</font></i>: 
<font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c35">i</font></span>)</span>,<font color="Maroon" title="c27">k2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green" title="E52">A81</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" title="E82">A162</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c36">y</font> = <font color="Maroon" title="c34">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E82">A162</font></i></a></i>;<br/><i><font color="Green" title="E84">A164</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c27">k2</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E80">A153</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E85">A165</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c35">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">Ck2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green" title="E58">A88</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" title="E82">A162</font></i></a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_16_2_1_2"/><i><font color="Green" title="E86">A166</font></i>: 
( <font color="Maroon" title="c26">Lma</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k2</font> &amp; <font color="Maroon" title="c28">Ck2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Ck2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green" title="E49">A78</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" title="E84">A164</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c28">Ck2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c35">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Ck2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c35">i</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" title="E85">A165</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c36">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Ck2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c35">i</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" title="E83">A163</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" title="E84">A164</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" title="E85">A165</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="c36">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c28">Ck2</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" title="E85">A165</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E87">A167</font></i>: 
( <font color="Maroon" title="c27">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">k</font> &amp; <font color="Maroon" title="c23">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green" title="E51">A80</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green" title="E55">A84</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2_16"><i><font color="Green" title="E75">A148</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" title="E81">A161</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c23">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">l</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">l</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" title="E45">A72</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c23">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Ck2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">ma</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green" title="E49">A78</font></i></a>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green" title="E51">A80</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" title="E86">A166</font></i></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c23">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c32">k</font></span>)</span> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c23">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</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" title="E84">A164</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" title="E87">A167</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E76">A149</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="E75">A168</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E76">A169</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c32">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_17"/><i><font color="Green" title="E77">A170</font></i>: 
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font> &amp; <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E76">A169</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c33">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:1_1_5_1_2_1_1_2_17"/><i><font color="Green" title="E78">A171</font></i>: 
( <font color="Maroon" title="c33">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h1</font> &amp; <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">n1</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_17"/><i><font color="Green" title="E79">A172</font></i>: 
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c33">n1</font>,<font color="Maroon" title="c16">k1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">n1</font> &amp; <font color="Maroon" title="c33">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">l1</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green" title="E36">A53</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E78">A171</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c34">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:1_1_5_1_2_1_1_2_17"/><i><font color="Green" title="E80">A173</font></i>: 
( <font color="Maroon" title="c34">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c34">n2</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" title="E77">A170</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c35">i1</font>, <font color="Maroon" title="c36">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:1_1_5_1_2_1_1_2_17"/><i><font color="Green" title="E81">A174</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c35">i1</font>,<font color="Maroon" title="c36">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c34">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c35">i1</font>,<font color="Maroon" title="c36">i2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E80">A173</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c37">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">i1</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_17"/><i><font color="Green" title="E82">A175</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c37">L</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_17"/><i><font color="Green" title="E83">A176</font></i>: 
( <font color="Maroon" title="c35">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c36">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E56">A85</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E81">A174</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_17"/><i><font color="Green" title="E84">A177</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c37">L</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">i2</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" title="E82">A175</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E83">A176</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c34">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">L</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">i2</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" title="E81">A174</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E83">A176</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c34">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">L</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c34">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E80">A173</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E82">A175</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E83">A176</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E84">A177</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">L</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="E85">A178</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">i1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E83">A176</font></i></a></i>;<br/>
<a NAME="E20:1_1_5_1_2_1_1_2_17"/>
<font color="Maroon" title="c33">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green" title="E57">A86</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E78">A171</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c33">n1</font>,<font color="Maroon" title="c16">k1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green" title="E37">A54</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_17"/><b>then </b>
( <font color="Maroon" title="c35">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">n1</font> &amp; 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="E79">A172</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E80">A173</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E81">A174</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_17"/><b>then </b>
( <font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c12">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">mi</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" title="E79">A172</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2_17"><i><font color="Green" title="E85">A178</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E76">A179</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c32">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_18"/><i><font color="Green" title="E77">A180</font></i>: 
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font> &amp; <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E76">A179</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c33">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:1_1_5_1_2_1_1_2_18"/><i><font color="Green" title="E78">A181</font></i>: 
( <font color="Maroon" title="c33">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font> &amp; <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">n1</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_18"/><i><font color="Green" title="E79">A182</font></i>: 
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c33">n1</font></span>)</span>,<font color="Maroon" title="c27">k2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">n1</font> &amp; <font color="Maroon" title="c33">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l2</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green" title="E52">A81</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E78">A181</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c34">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:1_1_5_1_2_1_1_2_18"/><i><font color="Green" title="E80">A183</font></i>: 
( <font color="Maroon" title="c34">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c34">n2</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" title="E77">A180</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c35">i1</font>, <font color="Maroon" title="c36">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:1_1_5_1_2_1_1_2_18"/><i><font color="Green" title="E81">A184</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c35">i1</font>,<font color="Maroon" title="c36">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c34">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c35">i1</font>,<font color="Maroon" title="c36">i2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E80">A183</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c37">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">i1</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_18"/><i><font color="Green" title="E82">A185</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c37">L</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_18"/><i><font color="Green" title="E83">A186</font></i>: 
( <font color="Maroon" title="c35">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c36">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E56">A85</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E81">A184</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_18"/><i><font color="Green" title="E84">A187</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c37">L</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">i2</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" title="E82">A185</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E83">A186</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c34">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">L</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">i2</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" title="E81">A184</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E83">A186</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c34">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">L</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c34">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E80">A183</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E82">A185</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E83">A186</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E84">A187</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">L</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="E85">A188</font></i>: 
<font color="Maroon" title="c35">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E83">A186</font></i></a></i>;<br/>
<a NAME="E20:1_1_5_1_2_1_1_2_18"/>
<font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c33">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green" title="E58">A88</font></i></a>, <a class="txt" href="goboard4.html#E5:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E78">A181</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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c33">n1</font></span>)</span>,<font color="Maroon" title="c27">k2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green" title="E51">A80</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_18"/><b>then </b>
<font color="Maroon" title="c35">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c33">n1</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" title="E79">A182</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E80">A183</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1_5_1_2_1_1_2_18"><i><font color="Green" title="E81">A184</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_18"/><b>then </b>
<font color="Maroon" title="c33">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E85">A188</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">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" title="E79">A182</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><b>then </b><i><font color="Green" title="E76">A189</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="goboard4.html#E51:1_1_5_1_2_1_1_2"><i><font color="Green" title="E73">A146</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E75">A168</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span>

;<br/><a NAME="E56:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E77">A190</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</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" title="c32">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_20"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c33">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_5_1_2_1_1_2_20"/><i><font color="Green" title="E78">A191</font></i>: 
( <font color="Maroon" title="c33">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> &amp; <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_20"/>
( <font color="Maroon" title="c33">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> &amp; <font color="Maroon" title="c33">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c33">n</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" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green" title="E54">A83</font></i></a>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2_20"><i><font color="Green" title="E78">A191</font></i></a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_20"/><b>then </b>
( <font color="Maroon" title="c24">w</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c33">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c33">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c33">n</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" title="E45">A72</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:1_1_5_1_2_1_1_2_20"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</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" title="E78">A191</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E78">A192</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="goboard4.html#E55:1_1_5_1_2_1_1_2"><i><font color="Green" title="E76">A189</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><a NAME="E58:1_1_5_1_2_1_1_2"/><i><font color="Green" title="E79">A193</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E59:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E80">A194</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="goboard1.html#D10" title="GOBOARD1:def.10">GOBOARD1:def 10</a></i>;<br/><a NAME="E60:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E81">A195</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</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" title="E70">A133</font></i></a>, <a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2"><i><font color="Green" title="E71">A134</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><i><font color="Green" title="E82">A196</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E80">A194</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="goboard4.html#E52:1_1_5_1_2_1_1_2"><i><font color="Green" title="E74">A147</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" title="E83">A197</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>7</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> );<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green" title="E84">A198</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green" title="E85">A199</font></i>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">m</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" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="goboard1.html#T22" title="GOBOARD1:th.22">GOBOARD1:22</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c32">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E86">A200</font></i>: 
( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c32">m</font>] &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">m</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">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" title="E84">A198</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" title="E85">A199</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c33">p</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c32">m</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><div><i><font color="Green" title="E87">A201</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" title="E88">A202</font></i>: 
<font color="Maroon" title="c32">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_2"/>
<font color="Maroon" title="c32">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E86">A200</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c33">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c33">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green" title="E13">A12</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" title="E86">A200</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" title="E88">A202</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">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" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">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" title="E38">A65</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c34">ll</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c32">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c35">t</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E88">A203</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">ll</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c35">t</font> holds <br/><font color="Maroon" title="c35">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1" title="GOBOARD2:sch.1">GOBOARD2:sch 1</a>();<br/></i><a NAME="E13:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green" title="E89">A204</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c35">t</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E90">A205</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c35">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c34">ll</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" title="E88">A203</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E91">A206</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c35">t</font> holds <br/><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="c36">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_3"/>
<font color="Maroon" title="c36">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c35">t</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">n</font> &amp; <font color="Maroon" title="c36">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">ll</font> &amp; <font color="Maroon" title="c36">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c32">m</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" title="E90">A205</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c32">m</font> &amp; <font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c36">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c34">ll</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c36">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="E92">A207</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c35">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="c36">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c36">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c35">t</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c37">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E93">A208</font></i>: 
( <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c35">t</font> &amp; <font color="Maroon" title="c35">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">y</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">x</font> &amp; <font color="Maroon" title="c37">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">ll</font> &amp; <font color="Maroon" title="c37">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c32">m</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" title="E90">A205</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" title="E93">A208</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c32">m</font> &amp; <font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c37">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c34">ll</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c37">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="c36">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E88">A203</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" title="E93">A208</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c36">t</font> = <font color="Maroon" title="c35">t</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><b>set </b><font color="Maroon" title="c37">D</font> =  <a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green" title="E93">A209</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</font></i></a>, <a class="ref" href="goboard1.html#T26" title="GOBOARD1:th.26">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" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">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" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green" title="E94">A210</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_1"/>
<font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E87">A201</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green" title="E95">A211</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">t</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" title="E88">A203</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c34">ll</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" title="E88">A203</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E96">A212</font></i>: 
<font color="Maroon" title="c36">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E88">A203</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" title="E89">A204</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" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</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" title="E93">A209</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" title="E94">A210</font></i></a>, <a class="ref" href="goboard1.html#T30" title="GOBOARD1:th.30">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" title="E97">A213</font></i>: 
<font color="Maroon" title="c36">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green" title="E15">A14</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" title="E86">A200</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" title="E96">A212</font></i></a>, <a class="ref" href="goboard1.html#T48" title="GOBOARD1:th.48">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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c34">ll</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" title="E88">A203</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" title="E95">A211</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>then </b><font color="Maroon" title="c36">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">t</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c34">ll</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" title="E88">A203</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" title="E89">A204</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E98">A214</font></i>: 
<font color="Maroon" title="c36">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</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" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</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" title="E93">A209</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" title="E94">A210</font></i></a>, <a class="ref" href="goboard1.html#T30" title="GOBOARD1:th.30">GOBOARD1:30</a></i>;<br/><a NAME="E31:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green" title="E99">A215</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></i>;<br/><div><i><font color="Green" title="E100">A216</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" title="c38">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E101">A217</font></i>: 
<font color="Maroon" title="c38">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c36">t</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b>
<font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E91">A206</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c39">i1</font>, <font color="Maroon" title="c40">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E102">A218</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c39">i1</font>,<font color="Maroon" title="c40">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c39">i1</font>,<font color="Maroon" title="c40">i2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon" title="c41">i1</font> = <font color="Maroon" title="c39">i1</font>;<br/><b>reconsider </b><font color="Maroon" title="c42">Ci2</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c40">i2</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E103">A219</font></i>: 
( <font color="Maroon" title="c41">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c40">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <font color="Maroon" title="c36">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c38">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font></span>)</span> &amp; <font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E56">A85</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" title="E88">A203</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" title="E91">A206</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" title="E101">A217</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" title="E102">A218</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Ci2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/><b>then </b><i><font color="Green" title="E104">A220</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">Ci2</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c42">Ci2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c41">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Ci2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">i1</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" title="E103">A219</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c42">Ci2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c41">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c41">i1</font>,<font color="Maroon" title="c40">i2</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c40">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c40">i2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c40">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E103">A219</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="E105">A221</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c40">i2</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" title="E102">A218</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" title="E103">A219</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" title="E104">A220</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="c40">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E106">A222</font></i>: 
( <font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">n</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" title="E86">A200</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" title="E90">A205</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" title="E101">A217</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" title="E103">A219</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" title="E105">A221</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">m</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" title="E106">A222</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c32">m</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="XXREAL_0:pred.1">&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" title="E106">A223</font></i>: 
( 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c40">i2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c40">i2</font> &amp; <font color="Maroon" title="c40">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E103">A219</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E107">A224</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c40">i2</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c43">l</font> = <font color="Maroon" title="c40">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E106">A223</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>take </b><font color="Maroon" title="c44">l</font> = <font color="Maroon" title="c43">l</font>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_1_6"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c40">i2</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" title="E107">A224</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E108">A225</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">l</font> &amp; <font color="Maroon" title="c44">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c44">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">i2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</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" title="E93">A209</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" title="E106">A223</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">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" title="c44">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c41">i1</font>,<font color="Maroon" title="c44">l</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</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" title="E82">A196</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" title="E99">A215</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" title="E103">A219</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="c36">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c38">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c41">i1</font>,<font color="Maroon" title="c44">l</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</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" title="E93">A209</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" title="E102">A218</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" title="E103">A219</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" title="E108">A225</font></i></a>, <a class="ref" href="goboard1.html#T32" title="GOBOARD1:th.32">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" title="c38">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E101">A226</font></i>: 
( <font color="Maroon" title="c38">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c36">t</font> &amp; <font color="Maroon" title="c38">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c36">t</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b><i><font color="Green" title="E102">A227</font></i>: 
( <font color="Maroon" title="c36">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c38">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font></span>)</span> &amp; <font color="Maroon" title="c36">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c38">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E88">A203</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c39">i1</font>, <font color="Maroon" title="c40">i2</font>, <font color="Maroon" title="c41">j1</font>, <font color="Maroon" title="c42">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E103">A228</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c39">i1</font>,<font color="Maroon" title="c40">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c41">j1</font>,<font color="Maroon" title="c42">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c36">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c38">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c39">i1</font>,<font color="Maroon" title="c40">i2</font> &amp; <font color="Maroon" title="c36">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c41">j1</font>,<font color="Maroon" title="c42">j2</font> )
 ;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b><i><font color="Green" title="E104">A229</font></i>: 
( <font color="Maroon" title="c39">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c40">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c41">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c42">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E93">A209</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" title="E99">A215</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c40">i2</font> &amp; <font color="Maroon" title="c40">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">j2</font> &amp; <font color="Maroon" title="c42">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E105">A230</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c39">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c40">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c41">j1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c42">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c40">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <font color="Maroon" title="c42">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</font></i></a>, <a class="txt" href="goboard4.html#E61:1_1_5_1_2_1_1_2"><i><font color="Green" title="E82">A196</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" title="E102">A227</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" title="E103">A228</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" title="E104">A229</font></i></a>, <a class="ref" href="goboard1.html#T32" title="GOBOARD1:th.32">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" title="E106">A231</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c39">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c40">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c41">j1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c42">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</font></i></a>, <a class="txt" href="goboard4.html#E61:1_1_5_1_2_1_1_2"><i><font color="Green" title="E82">A196</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" title="E104">A229</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_1_7"/>
( <font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c32">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c38">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E91">A206</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" title="E101">A226</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" title="E104">A229</font></i></a></i>;<br/><b>hence </b>1 = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c39">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c41">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c40">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c42">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E11">A10</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" title="E105">A230</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" title="E106">A231</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c39">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c41">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c40">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c42">j2</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" title="E101">A232</font></i>: 
<font color="Maroon" title="c36">t</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</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" title="E100">A216</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">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" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</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" title="E15">A14</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1_5_1_2_1_1_2"><i><font color="Green" title="E68">A119</font></i></a>, <a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2"><i><font color="Green" title="E69">A120</font></i></a>, <a class="txt" href="goboard4.html#E50:1_1_5_1_2_1_1_2"><i><font color="Green" title="E72">A135</font></i></a>, <a class="txt" href="goboard4.html#E58:1_1_5_1_2_1_1_2"><i><font color="Green" title="E79">A193</font></i></a>, <a class="txt" href="goboard4.html#E59:1_1_5_1_2_1_1_2"><i><font color="Green" title="E80">A194</font></i></a>, <a class="txt" href="goboard4.html#E60:1_1_5_1_2_1_1_2"><i><font color="Green" title="E81">A195</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" title="E83">A197</font></i></a>, <a class="ref" href="goboard1.html#T37" title="GOBOARD1:th.37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41" title="GOBOARD1:th.41">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" title="E102">A233</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c36">t</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</font></i></a>, <a class="txt" href="goboard4.html#E47:1_1_5_1_2_1_1_2"><i><font color="Green" title="E70">A133</font></i></a>, <a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2"><i><font color="Green" title="E71">A134</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" title="E93">A209</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" title="E95">A211</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" title="E97">A213</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" title="E98">A214</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" title="E101">A232</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c38">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c36">t</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" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c36">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E92">A207</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">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" title="c38">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E102">A233</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E57:1_1_5_1_2_1_1_2"><i><font color="Green" title="E78">A192</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" title="E83">A234</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c32">D</font> =  <a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2"/><i><font color="Green" title="E84">A235</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="goboard4.html#E52:1_1_5_1_2_1_1_2"><i><font color="Green" title="E74">A147</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" title="E85">A236</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>consider </b><font color="Maroon" title="c33">t</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E86">A237</font></i>: 
( <font color="Maroon" title="c33">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <font color="Maroon" title="c33">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">t</font> &amp; <font color="Maroon" title="c33">t</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green" title="E13">A12</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" title="E84">A235</font></i></a>, <a class="ref" href="goboard1.html#T51" title="GOBOARD1:th.51">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" title="E87">A238</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</font></i></a>, <a class="ref" href="goboard1.html#T26" title="GOBOARD1:th.26">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" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1_5_1_2_1_1_2"><i><font color="Green" title="E68">A119</font></i></a>, <a class="txt" href="goboard4.html#E46:1_1_5_1_2_1_1_2"><i><font color="Green" title="E69">A120</font></i></a>, <a class="txt" href="goboard4.html#E50:1_1_5_1_2_1_1_2"><i><font color="Green" title="E72">A135</font></i></a>, <a class="txt" href="goboard4.html#E58:1_1_5_1_2_1_1_2"><i><font color="Green" title="E79">A193</font></i></a>, <a class="txt" href="goboard4.html#E59:1_1_5_1_2_1_1_2"><i><font color="Green" title="E80">A194</font></i></a>, <a class="txt" href="goboard4.html#E60:1_1_5_1_2_1_1_2"><i><font color="Green" title="E81">A195</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" title="E85">A236</font></i></a>, <a class="ref" href="goboard1.html#T37" title="GOBOARD1:th.37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41" title="GOBOARD1:th.41">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" title="E88">A239</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">t</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</font></i></a>, <a class="txt" href="goboard4.html#E47:1_1_5_1_2_1_1_2"><i><font color="Green" title="E70">A133</font></i></a>, <a class="txt" href="goboard4.html#E49:1_1_5_1_2_1_1_2"><i><font color="Green" title="E71">A134</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" title="E86">A237</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" title="E87">A238</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c34">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">t</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" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E86">A237</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">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" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E88">A239</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E57:1_1_5_1_2_1_1_2"><i><font color="Green" title="E78">A192</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" title="E85">A240</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E86">A241</font></i>: 
<font color="Maroon" title="c25">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green" title="E61">A93</font></i></a>, <a class="txt" href="goboard4.html#E40:1_1_5_1_2_1_1_2"><i><font color="Green" title="E66">A104</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c33">t</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E87">A242</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font> &amp; <font color="Maroon" title="c33">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c33">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">t</font> &amp; <font color="Maroon" title="c33">t</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</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" title="E83">A234</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" title="E85">A240</font></i></a>, <a class="ref" href="goboard1.html#T52" title="GOBOARD1:th.52">GOBOARD1:52</a></i>;<br/><b>consider </b><font color="Maroon" title="c34">tt</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E88">A243</font></i>: 
( <font color="Maroon" title="c34">tt</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <font color="Maroon" title="c34">tt</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c34">tt</font> &amp; <font color="Maroon" title="c34">tt</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">t</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</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" title="E84">A235</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" title="E87">A242</font></i></a>, <a class="ref" href="goboard1.html#T51" title="GOBOARD1:th.51">GOBOARD1:51</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>7</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font></span>)</span> );<br/><div><i><font color="Green" title="E89">A244</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" title="c35">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</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" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c36">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E90">A245</font></i>: 
( <font color="Maroon" title="c36">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><b>take </b><font color="Maroon" title="c37">n</font> = <font color="Maroon" title="c36">n</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" title="c37">n</font>]
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</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" title="E90">A245</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c13">ma</font></span>)</span> );<br/><div><i><font color="Green" title="E90">A246</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" title="c35">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c13">ma</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" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c13">ma</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c36">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E91">A247</font></i>: 
( <font color="Maroon" title="c36">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><b>take </b><font color="Maroon" title="c37">n</font> = <font color="Maroon" title="c36">n</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" title="c37">n</font>]
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</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" title="E91">A247</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c35">pf</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E91">A248</font></i>: 
( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c35">pf</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c35">pf</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">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" title="E89">A244</font></i></a>);<br/></i><b>consider </b><font color="Maroon" title="c36">pl</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E92">A249</font></i>: 
( <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c36">pl</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c36">pl</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">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" title="E90">A246</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" title="E93">A250</font></i>: 
<font color="Maroon" title="c36">pl</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c35">pf</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green" title="E40">A67</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" title="E91">A248</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" title="E92">A249</font></i></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">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" title="E94">A251</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">pf</font> &amp; <font color="Maroon" title="c35">pf</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</font> &amp; <font color="Maroon" title="c36">pl</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E91">A248</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" title="E92">A249</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c35">pf</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">pl</font> or <font color="Maroon" title="c35">pf</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c36">pl</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" title="E93">A250</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c35">pf</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">pl</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" title="c35">pf</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E94">A251</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E94">A251</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="c35">pf</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c36">pl</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" title="c36">pl</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E94">A251</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E94">A251</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E95">A252</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="goboard1.html#T38" title="GOBOARD1:th.38">GOBOARD1:38</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c37">lg</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">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" title="c37">lg</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">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" title="E96">A253</font></i>: 
( <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c37">lg</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E95">A252</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c38">C</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>, <font color="Maroon" title="c39">Li</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</font>, <font color="Maroon" title="c40">La</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c13">ma</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E97">A254</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green" title="E30">A32</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green" title="E31">A33</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/><b>then </b><i><font color="Green" title="E98">A255</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">C</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><b>consider </b><font color="Maroon" title="c41">K1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E99">A256</font></i>: 
( <font color="Maroon" title="c41">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Li</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Li</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c41">K1</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" title="E91">A248</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E100">A257</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Li</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Li</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c42">CK1</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c41">K1</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E101">A258</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Li</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E102">A259</font></i>: 
<font color="Maroon" title="c39">Li</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c41">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Li</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">K1</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" title="E99">A256</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="E103">A260</font></i>: 
( <font color="Maroon" title="c41">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">mi</font>,<font color="Maroon" title="c41">K1</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E104">A261</font></i>: 
<font color="Maroon" title="c35">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon" title="c43">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E105">A262</font></i>: 
( <font color="Maroon" title="c43">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">C</font> &amp; <font color="Maroon" title="c38">C</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c43">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="c38">C</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c43">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i2</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" title="E105">A262</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="E106">A263</font></i>: 
( <font color="Maroon" title="c43">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c43">i2</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E98">A255</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" title="E105">A262</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="E107">A264</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c43">i2</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E98">A255</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" title="E105">A262</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">mi</font>,<font color="Maroon" title="c41">K1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E108">A265</font></i>: 
( <font color="Maroon" title="c43">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">K1</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" title="E103">A260</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" title="E104">A261</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" title="E106">A263</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" title="E107">A264</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>consider </b><font color="Maroon" title="c44">n1</font>, <font color="Maroon" title="c45">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E109">A266</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c44">n1</font>,<font color="Maroon" title="c45">n2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c44">n1</font>,<font color="Maroon" title="c45">n2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</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" title="E96">A253</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">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" title="E110">A267</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c44">n1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">mi</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c45">n2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</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" title="E96">A253</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" title="E106">A263</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" title="E107">A264</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" title="E108">A265</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" title="E109">A266</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">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" title="E111">A268</font></i>: 
( <font color="Maroon" title="c44">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c45">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E56">A85</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" title="E109">A266</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">n1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">mi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c45">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> ) or (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">n2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c44">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</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" title="E110">A267</font></i></a>, <a class="ref" href="goboard1.html#T2" title="GOBOARD1:th.2">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" title="E112">A269</font></i>: 
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">n1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">mi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c45">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 ;<br/><div class="add"><i><font color="Green" title="E113">A270</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">C</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">C</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c38">C</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c44">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">n1</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" title="E111">A268</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">C</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c44">n1</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" title="E109">A266</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" title="E111">A268</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" title="E112">A269</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</font> &amp; 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green" title="E31">A33</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" title="E96">A253</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" title="E111">A268</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" title="E113">A270</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">lg</font> &amp; <font color="Maroon" title="c37">lg</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green" title="E31">A33</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" title="E96">A253</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" title="E97">A254</font></i></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">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" title="E112">A271</font></i>: 
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">n2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c44">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Li</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E113">A272</font></i>: 
<font color="Maroon" title="c45">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">Li</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" title="E111">A268</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c39">Li</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c45">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Li</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">n2</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Li</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c45">n2</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" title="E109">A266</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" title="E111">A268</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" title="E112">A271</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c39">Li</font> &amp; 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="E113">A272</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">lg</font> &amp; <font color="Maroon" title="c37">lg</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E91">A248</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" title="E96">A253</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" title="E104">A261</font></i></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">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" title="E104">A273</font></i>: 
<font color="Maroon" title="c35">pf</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E94">A251</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c43">K2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E105">A274</font></i>: 
( <font color="Maroon" title="c43">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">Lma</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">pl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c43">K2</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" title="E92">A249</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E106">A275</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">Lma</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">Lma</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c44">CK2</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c43">K2</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E107">A276</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">Lma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E108">A277</font></i>: 
<font color="Maroon" title="c26">Lma</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c43">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">K2</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" title="E105">A274</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="E109">A278</font></i>: 
( <font color="Maroon" title="c43">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">pl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">ma</font>,<font color="Maroon" title="c43">K2</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E110">A279</font></i>: 
<font color="Maroon" title="c36">pl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon" title="c45">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E111">A280</font></i>: 
( <font color="Maroon" title="c45">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">C</font> &amp; <font color="Maroon" title="c38">C</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c45">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/><i><font color="Green" title="E112">A281</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">C</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">C</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c38">C</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c45">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">i2</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" title="E111">A280</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="E113">A282</font></i>: 
( <font color="Maroon" title="c45">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c45">i2</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E111">A280</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" title="E112">A281</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="E114">A283</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c45">i2</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E111">A280</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" title="E112">A281</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">ma</font>,<font color="Maroon" title="c43">K2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E115">A284</font></i>: 
( <font color="Maroon" title="c45">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">K2</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" title="E109">A278</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" title="E110">A279</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" title="E113">A282</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" title="E114">A283</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>consider </b><font color="Maroon" title="c46">n1</font>, <font color="Maroon" title="c47">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E116">A285</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c46">n1</font>,<font color="Maroon" title="c47">n2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c46">n1</font>,<font color="Maroon" title="c47">n2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</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" title="E96">A253</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">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" title="E117">A286</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c46">n1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c13">ma</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">n2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</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" title="E96">A253</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" title="E113">A282</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" title="E114">A283</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" title="E115">A284</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" title="E116">A285</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">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" title="E118">A287</font></i>: 
( <font color="Maroon" title="c46">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c47">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E56">A85</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" title="E116">A285</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">n1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c13">ma</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c47">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> ) or (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c46">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</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" title="E117">A286</font></i></a>, <a class="ref" href="goboard1.html#T2" title="GOBOARD1:th.2">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" title="E119">A288</font></i>: 
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">n1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c13">ma</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c47">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 ;<br/><div class="add"><i><font color="Green" title="E120">A289</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">C</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">C</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c38">C</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c46">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">n1</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" title="E118">A287</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">C</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c46">n1</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" title="E116">A285</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" title="E118">A287</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" title="E119">A288</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</font> &amp; 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green" title="E31">A33</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" title="E96">A253</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" title="E118">A287</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" title="E120">A289</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">lg</font> &amp; <font color="Maroon" title="c37">lg</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green" title="E31">A33</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" title="E96">A253</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" title="E97">A254</font></i></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">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" title="E119">A290</font></i>: 
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c46">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">Lma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E120">A291</font></i>: 
<font color="Maroon" title="c47">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">Lma</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" title="E118">A287</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c40">La</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">n2</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">n2</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" title="E116">A285</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" title="E118">A287</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" title="E119">A290</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">Lma</font> &amp; 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="E120">A291</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">lg</font> &amp; <font color="Maroon" title="c37">lg</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E92">A249</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" title="E96">A253</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" title="E110">A279</font></i></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">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" title="E110">A292</font></i>: 
<font color="Maroon" title="c36">pl</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E94">A251</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> $1,<font color="Maroon" title="c41">K1</font>;<br/><b>consider </b><font color="Maroon" title="c45">h1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E111">A293</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">l1</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</font> holds <br/><font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1" title="GOBOARD2:sch.1">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" title="E112">A294</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> $1</span>)</span>,<font color="Maroon" title="c43">K2</font>;<br/><b>consider </b><font color="Maroon" title="c46">h2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E113">A295</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l2</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</font> holds <br/><font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1" title="GOBOARD2:sch.1">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" title="E114">A296</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><i><font color="Green" title="E115">A297</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" title="c47">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</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" title="E116">A298</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c47">n</font> &amp; <font color="Maroon" title="c47">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">l1</font> &amp; <font color="Maroon" title="c18">l1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font> &amp; <font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</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" title="E111">A293</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">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" title="c47">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="c47">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E116">A298</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</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" title="E116">A298</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E116">A299</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" title="c47">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c47">n</font> &amp; <font color="Maroon" title="c47">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l2</font> &amp; <font color="Maroon" title="c47">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c13">ma</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" title="E113">A295</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c13">ma</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">l2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E117">A300</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" title="c47">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</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" title="E118">A301</font></i>: 
( <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c47">n</font>,<font color="Maroon" title="c41">K1</font> &amp; <font color="Maroon" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</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" title="E111">A293</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" title="E115">A297</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c48">i</font> = <font color="Maroon" title="c47">n</font>;<br/><b>take </b><font color="Maroon" title="c49">K1</font> = <font color="Maroon" title="c41">K1</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c48">i</font>,<font color="Maroon" title="c49">K1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c48">i</font>,<font color="Maroon" title="c49">K1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</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" title="E118">A301</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E118">A302</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" title="c47">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</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" title="E119">A303</font></i>: 
( <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font></span>)</span>,<font color="Maroon" title="c43">K2</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</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" title="E113">A295</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" title="E116">A299</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c48">m</font> = <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font>;<br/><b>take </b><font color="Maroon" title="c49">K2</font> = <font color="Maroon" title="c43">K2</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c48">m</font>,<font color="Maroon" title="c49">K2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c48">m</font>,<font color="Maroon" title="c49">K2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</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" title="E119">A303</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E119">A304</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" title="c47">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</font> &amp; <font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</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" title="E120">A305</font></i>: 
( <font color="Maroon" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c47">n</font>,<font color="Maroon" title="c41">K1</font> &amp; <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c41">K1</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" title="E111">A293</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" title="E115">A297</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" title="E121">A306</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c47">n</font>,<font color="Maroon" title="c41">K1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c41">K1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon" title="c48">i1</font>, <font color="Maroon" title="c49">i2</font>, <font color="Maroon" title="c50">j1</font>, <font color="Maroon" title="c51">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c48">i1</font>,<font color="Maroon" title="c49">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c50">j1</font>,<font color="Maroon" title="c51">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c48">i1</font>,<font color="Maroon" title="c49">i2</font> &amp; <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c50">j1</font>,<font color="Maroon" title="c51">j2</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" title="c48">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">n</font> &amp; <font color="Maroon" title="c49">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">K1</font> &amp; <font color="Maroon" title="c50">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c51">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">K1</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E120">A305</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" title="E121">A306</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c48">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c50">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c49">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c51">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c47">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c47">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><i><font color="Green" title="E120">A307</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" title="c47">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</font> &amp; <font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</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" title="E121">A308</font></i>: 
( <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font></span>)</span>,<font color="Maroon" title="c43">K2</font> &amp; <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c43">K2</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E113">A295</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" title="E116">A299</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" title="E122">A309</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font></span>)</span>,<font color="Maroon" title="c43">K2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c43">K2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon" title="c48">i1</font>, <font color="Maroon" title="c49">i2</font>, <font color="Maroon" title="c50">j1</font>, <font color="Maroon" title="c51">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c48">i1</font>,<font color="Maroon" title="c49">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c50">j1</font>,<font color="Maroon" title="c51">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c48">i1</font>,<font color="Maroon" title="c49">i2</font> &amp; <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c50">j1</font>,<font color="Maroon" title="c51">j2</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" title="c48">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font> &amp; <font color="Maroon" title="c49">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">K2</font> &amp; <font color="Maroon" title="c50">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c51">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">K2</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E121">A308</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" title="E122">A309</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c48">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c50">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c49">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c51">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> implies for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">m</font></span>)</span> holds <br/><font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> );<br/><a NAME="E54:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green" title="E121">A310</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[0]
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="E122">A311</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c47">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E123">A312</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c47">k</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" title="E124">A313</font></i>: 
<font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c48">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E125">A314</font></i>: 
( <font color="Maroon" title="c48">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c48">m</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" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c47">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="E126">A315</font></i>: 
<font color="Maroon" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E127">A316</font></i>: 
<font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">ma1</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" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green" title="E55">A84</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">ma1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">ma1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">ma1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green" title="E45">A72</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</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" title="E127">A316</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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c39">Li</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green" title="E34">A51</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" title="E126">A315</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">m</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</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" title="E125">A314</font></i></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">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" title="c47">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c47">k</font> &amp; <font color="Maroon" title="c47">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</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" title="E124">A313</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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" title="E126">A317</font></i>: 
( 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c47">k</font> &amp; <font color="Maroon" title="c47">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E127">A318</font></i>: 
<font color="Maroon" title="c47">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c49">i1</font>, <font color="Maroon" title="c50">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E128">A319</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c49">i1</font>,<font color="Maroon" title="c50">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c49">i1</font>,<font color="Maroon" title="c50">i2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green" title="E61">A93</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c51">Li1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c49">i1</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E129">A320</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">Li1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c51">Li1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E130">A321</font></i>: 
( <font color="Maroon" title="c49">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c50">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E56">A85</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" title="E128">A319</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E131">A322</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">Li1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E132">A323</font></i>: 
<font color="Maroon" title="c51">Li1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">Li1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c50">i2</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" title="E129">A320</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" title="E130">A321</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">Li1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">i2</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" title="E128">A319</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" title="E130">A321</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E133">A324</font></i>: 
<font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">Li1</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" title="E129">A320</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" title="E130">A321</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" title="E131">A322</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E134">A325</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">i1</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" title="E123">A312</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" title="E126">A317</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" title="E130">A321</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>consider </b><font color="Maroon" title="c52">j1</font>, <font color="Maroon" title="c53">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E135">A326</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c52">j1</font>,<font color="Maroon" title="c53">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c52">j1</font>,<font color="Maroon" title="c53">j2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green" title="E61">A93</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" title="E124">A313</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c54">Lj1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">j1</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E136">A327</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c54">Lj1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c54">Lj1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E137">A328</font></i>: 
( <font color="Maroon" title="c52">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c53">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E56">A85</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" title="E135">A326</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E138">A329</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c54">Lj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c54">Lj1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c53">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c54">Lj1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">j2</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" title="E136">A327</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" title="E137">A328</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c54">Lj1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c53">j2</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" title="E135">A326</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" title="E137">A328</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E139">A330</font></i>: 
<font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c54">Lj1</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" title="E136">A327</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" title="E137">A328</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" title="E138">A329</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">i1</font> or <font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c49">i1</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" title="E134">A325</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="E140">A331</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">i1</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" title="E141">A332</font></i>: 
( <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font></span>)</span> &amp; <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green" title="E45">A72</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</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" title="E127">A318</font></i></a></i>;<br/><font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</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" title="E45">A72</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</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" title="E127">A318</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">k</font>
<b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</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" title="E127">A318</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c51">Li1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">i2</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" title="E128">A319</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" title="E130">A321</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" title="E132">A323</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E142">A333</font></i>: 
<font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c12">mi</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" title="E129">A320</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" title="E130">A321</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" title="E131">A322</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" title="E140">A331</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</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" title="E126">A317</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">ma1</font> &amp; <font color="Maroon" title="c14">ma1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green" title="E29">A31</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" title="E141">A332</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" title="E142">A333</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" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">ma1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">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" title="E143">A334</font></i>: 
<font color="Maroon" title="c14">ma1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E144">A335</font></i>: 
( <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c47">k</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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" title="E45">A72</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</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" title="E124">A313</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" title="c12">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> &amp; <font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E42">A69</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green" title="E26">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="E145">A336</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green" title="E29">A31</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" title="E143">A334</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" title="E144">A335</font></i></a>, <a class="ref" href="goboard1.html#T44" title="GOBOARD1:th.44">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" title="c48">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E125">A314</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" title="E145">A336</font></i></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">GOBOARD1:19</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">m</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" title="E140">A337</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c49">i1</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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c49">i1</font></span>)</span> or for <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">l</font></span>)</span> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E86">A241</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" title="E124">A313</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" title="E127">A318</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" title="E130">A321</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" title="E133">A324</font></i></a>, <a class="ref" href="goboard1.html#T43" title="GOBOARD1:th.43">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c49">i1</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" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">m</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" title="E125">A314</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" title="E130">A321</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" title="E140">A337</font></i></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">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" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">l</font></span>)</span> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E141">A338</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c52">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E137">A328</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" title="E139">A330</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" title="c52">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c49">i1</font> &amp; <font color="Maroon" title="c49">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c49">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c52">j1</font> &amp; <font color="Maroon" title="c52">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E141">A338</font></i></a>, <a class="ref" href="goboard1.html#T1" title="GOBOARD1:th.1">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" title="c52">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c49">i1</font> &amp; <font color="Maroon" title="c49">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c49">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">j1</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" title="E140">A337</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">m</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" title="E125">A314</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" title="E137">A328</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" title="E139">A330</font></i></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">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" title="c49">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c52">j1</font> &amp; <font color="Maroon" title="c52">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c12">mi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c52">j1</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" title="E140">A337</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">m</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" title="E125">A314</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" title="E137">A328</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" title="E139">A330</font></i></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">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" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">m</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" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">m</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" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">m</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" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">m</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" title="E123">A339</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">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" title="E121">A310</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" title="E122">A311</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" title="E124">A340</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E125">A341</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c47">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</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" title="E126">A342</font></i>: 
( <font color="Maroon" title="c47">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font> &amp; <font color="Maroon" title="c47">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</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" title="E125">A341</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c47">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">t</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" title="E88">A243</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c48">i1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E127">A343</font></i>: 
( <font color="Maroon" title="c48">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">x</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" title="E87">A242</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c49">n1</font>, <font color="Maroon" title="c50">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E128">A344</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c49">n1</font>,<font color="Maroon" title="c50">n2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c49">n1</font>,<font color="Maroon" title="c50">n2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green" title="E61">A93</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" title="E127">A343</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c51">Ln1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c49">n1</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E129">A345</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">Ln1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c51">Ln1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E130">A346</font></i>: 
( <font color="Maroon" title="c49">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c50">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E56">A85</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" title="E128">A344</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E131">A347</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">Ln1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c51">Ln1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">Ln1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c50">n2</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" title="E129">A345</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" title="E130">A346</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">Ln1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">n2</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" title="E128">A344</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" title="E130">A346</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">Ln1</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" title="E129">A345</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" title="E130">A346</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" title="E131">A347</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E132">A348</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">n1</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" title="E123">A339</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" title="E127">A343</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" title="E130">A346</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c52">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E133">A349</font></i>: 
( <font color="Maroon" title="c52">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</font> &amp; <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">x</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" title="E126">A342</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E134">A350</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E135">A351</font></i>: 
( <font color="Maroon" title="c47">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c52">i2</font>,<font color="Maroon" title="c41">K1</font> &amp; <font color="Maroon" title="c52">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</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" title="E111">A293</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" title="E115">A297</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" title="E133">A349</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c52">i2</font>,<font color="Maroon" title="c41">K1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E136">A352</font></i>: 
( <font color="Maroon" title="c52">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">n1</font> &amp; <font color="Maroon" title="c50">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">K1</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" title="E127">A343</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" title="E128">A344</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" title="E135">A351</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">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" title="c18">l1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">mi</font> &amp; <font color="Maroon" title="c52">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">l1</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" title="E111">A293</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" title="E133">A349</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" title="E134">A350</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">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" title="E132">A348</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" title="E136">A352</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>10</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> implies for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">m</font></span>)</span> holds <br/><font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font> );<br/><a NAME="E58:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green" title="E125">A353</font></i>: 
<font color="Maroon">S<sub>10</sub></font>[0]
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="E126">A354</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>10</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>10</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c47">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E127">A355</font></i>: 
<font color="Maroon">S<sub>10</sub></font>[<font color="Maroon" title="c47">k</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" title="E128">A356</font></i>: 
<font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c48">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E129">A357</font></i>: 
( <font color="Maroon" title="c48">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c48">m</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" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c47">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="E130">A358</font></i>: 
<font color="Maroon" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c23">l</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green" title="E55">A84</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E45">A72</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c39">Li</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green" title="E34">A51</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" title="E130">A358</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="c48">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green" title="E41">A68</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" title="E129">A357</font></i></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">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" title="c47">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="E130">A359</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c47">k</font> &amp; <font color="Maroon" title="c47">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green" title="E54">A83</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" title="E128">A356</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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" title="E131">A360</font></i>: 
( 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c47">k</font> &amp; <font color="Maroon" title="c47">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E132">A361</font></i>: 
<font color="Maroon" title="c47">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c49">i1</font>, <font color="Maroon" title="c50">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E133">A362</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c49">i1</font>,<font color="Maroon" title="c50">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c49">i1</font>,<font color="Maroon" title="c50">i2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green" title="E61">A93</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c51">Li1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c49">i1</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E134">A363</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">Li1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c51">Li1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E135">A364</font></i>: 
( <font color="Maroon" title="c49">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c50">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E56">A85</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" title="E133">A362</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E136">A365</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">Li1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c51">Li1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">Li1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c50">i2</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" title="E134">A363</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" title="E135">A364</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">Li1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">i2</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" title="E133">A362</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" title="E135">A364</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E137">A366</font></i>: 
<font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">Li1</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" title="E134">A363</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" title="E135">A364</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" title="E136">A365</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E138">A367</font></i>: 
<font color="Maroon" title="c49">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</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" title="E127">A355</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" title="E131">A360</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" title="E135">A364</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>consider </b><font color="Maroon" title="c52">j1</font>, <font color="Maroon" title="c53">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E139">A368</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c52">j1</font>,<font color="Maroon" title="c53">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c52">j1</font>,<font color="Maroon" title="c53">j2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green" title="E61">A93</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" title="E128">A356</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c54">Lj1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">j1</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E140">A369</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c54">Lj1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c54">Lj1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E141">A370</font></i>: 
( <font color="Maroon" title="c52">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c53">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E56">A85</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" title="E139">A368</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E142">A371</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c54">Lj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c54">Lj1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c53">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c54">Lj1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">j2</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" title="E140">A369</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" title="E141">A370</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c54">Lj1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c53">j2</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" title="E139">A368</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" title="E141">A370</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E143">A372</font></i>: 
<font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c54">Lj1</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" title="E140">A369</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" title="E141">A370</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" title="E142">A371</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E144">A373</font></i>: 
<font color="Maroon" title="c52">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">m</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" title="E129">A357</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" title="E141">A370</font></i></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">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" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">i1</font> or <font color="Maroon" title="c49">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">ma</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" title="E138">A367</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="E145">A374</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">i1</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" title="E146">A375</font></i>: 
( <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font></span>)</span> &amp; <font color="Maroon" title="c24">w</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font></span>)</span> &amp; <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c21">mi1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c47">k</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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" title="E45">A72</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green" title="E48">A77</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" title="E128">A356</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" title="E132">A361</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" title="E147">A376</font></i>: 
<font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</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" title="E131">A360</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="c14">ma1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green" title="E40">A67</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" title="E137">A366</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" title="E145">A374</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" title="E146">A375</font></i></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">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" title="c14">ma1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</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" title="E147">A376</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="E148">A377</font></i>: 
<font color="Maroon" title="c21">mi1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green" title="E44">A71</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" title="E137">A366</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" title="E145">A374</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" title="E146">A375</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" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">mi1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green" title="E47">A76</font></i></a>, <a class="txt" href="goboard4.html#E38:1_1_5_1_2_1_1_2"><i><font color="Green" title="E64">A100</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" title="E131">A360</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="c24">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c47">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">mi1</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" title="E148">A377</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">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" title="E47">A76</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" title="E130">A359</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E145">A378</font></i>: 
<font color="Maroon" title="c49">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">ma</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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c49">i1</font></span>)</span> or for <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">l</font></span>)</span> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E86">A241</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" title="E128">A356</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" title="E132">A361</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" title="E135">A364</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" title="E137">A366</font></i></a>, <a class="ref" href="goboard1.html#T43" title="GOBOARD1:th.43">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c49">i1</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" title="c48">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</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" title="E129">A357</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" title="E135">A364</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" title="E145">A378</font></i></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">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" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">l</font></span>)</span> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E146">A379</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c52">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E141">A370</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" title="E143">A372</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" title="c52">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c49">i1</font> &amp; <font color="Maroon" title="c49">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c49">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c52">j1</font> &amp; <font color="Maroon" title="c52">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E146">A379</font></i></a>, <a class="ref" href="goboard1.html#T1" title="GOBOARD1:th.1">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" title="c52">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c49">i1</font> &amp; <font color="Maroon" title="c49">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c48">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</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" title="E144">A373</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" title="E145">A378</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="c49">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c52">j1</font> &amp; <font color="Maroon" title="c52">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c48">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</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" title="E144">A373</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" title="E145">A378</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="c48">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</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" title="c48">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</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" title="c48">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</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" title="c48">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</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" title="E127">A380</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>10</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">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" title="E125">A353</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" title="E126">A354</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" title="E128">A381</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E129">A382</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c47">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</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" title="E130">A383</font></i>: 
( <font color="Maroon" title="c47">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font> &amp; <font color="Maroon" title="c47">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</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" title="E129">A382</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c47">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">t</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" title="E88">A243</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c48">i1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E131">A384</font></i>: 
( <font color="Maroon" title="c48">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">f</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">x</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" title="E87">A242</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c49">n1</font>, <font color="Maroon" title="c50">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E132">A385</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c49">n1</font>,<font color="Maroon" title="c50">n2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c49">n1</font>,<font color="Maroon" title="c50">n2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1_5_1_2_1_1_2"><i><font color="Green" title="E61">A93</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" title="E131">A384</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c51">Ln1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c49">n1</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E133">A386</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">Ln1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c51">Ln1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E134">A387</font></i>: 
( <font color="Maroon" title="c49">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c50">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E56">A85</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" title="E132">A385</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E135">A388</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">Ln1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c51">Ln1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">Ln1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c50">n2</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" title="E133">A386</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" title="E134">A387</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">Ln1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">n2</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" title="E132">A385</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" title="E134">A387</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c25">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">Ln1</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" title="E133">A386</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" title="E134">A387</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" title="E135">A388</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E136">A389</font></i>: 
<font color="Maroon" title="c49">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ma</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" title="E127">A380</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" title="E131">A384</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" title="E134">A387</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c52">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E137">A390</font></i>: 
( <font color="Maroon" title="c52">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</font> &amp; <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">x</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" title="E130">A383</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E138">A391</font></i>: 
( <font color="Maroon" title="c47">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">i2</font></span>)</span>,<font color="Maroon" title="c43">K2</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</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" title="E113">A295</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" title="E116">A299</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" title="E137">A390</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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">i2</font></span>)</span>,<font color="Maroon" title="c43">K2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E139">A392</font></i>: 
( <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">n1</font> &amp; <font color="Maroon" title="c50">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">K2</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" title="E131">A384</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" title="E132">A385</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" title="E138">A391</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">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" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c52">i2</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" title="E137">A390</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c52">i2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E136">A389</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" title="E139">A392</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">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" title="c35">pf</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">pl</font> or <font color="Maroon" title="c36">pl</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">pf</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" title="E93">A250</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="E129">A393</font></i>: 
<font color="Maroon" title="c35">pf</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">pl</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c47">F1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</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" title="c35">pf</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</font> &amp; <font color="Maroon" title="c36">pl</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E129">A393</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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" title="c35">pf</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c48">LL</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c36">pl</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c35">pf</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c49">w1</font> = <font color="Maroon" title="c35">pf</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E94">A251</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">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" title="E130">A394</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="c50">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E131">A395</font></i>: 
<font color="Maroon" title="c50">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</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" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">pf</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green" title="E20">A19</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" title="E91">A248</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">w1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c50">n</font> &amp; <font color="Maroon" title="c50">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">LL</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" title="E131">A395</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="E132">A396</font></i>: 
( 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c50">n</font> &amp; <font color="Maroon" title="c50">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c35">pf</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">LL</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c35">pf</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="c50">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">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" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c50">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c36">pl</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" title="E132">A396</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c50">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c50">n</font></span>)</span> &amp; <font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c50">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E92">A249</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>11</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> $1</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" title="E131">A397</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>11</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">p</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c50">F</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E132">A398</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">LL</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> holds <br/><font color="Maroon">S<sub>11</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font>] ) )
 <b>from </b><i><a class="ref" href="goboard1.html#S1" title="GOBOARD1:sch.1">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" title="E131">A397</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" title="E133">A399</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c51">FF</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</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" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</font> &amp; <font color="Maroon" title="c36">pl</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E129">A393</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="E134">A400</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">LL</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">LL</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" title="E132">A398</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="c52">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c52">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</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" title="E135">A401</font></i>: 
( <font color="Maroon" title="c52">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">n</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" title="E132">A398</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" title="E133">A399</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" title="E136">A402</font></i>: 
( <font color="Maroon" title="c49">w1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">n</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" title="E130">A394</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c53">w</font> = <font color="Maroon" title="c49">w1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>consider </b><font color="Maroon" title="c54">i</font>, <font color="Maroon" title="c55">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E137">A403</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">i</font>,<font color="Maroon" title="c55">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">i</font>,<font color="Maroon" title="c55">j</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</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" title="E136">A402</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon" title="c56">i</font> = <font color="Maroon" title="c54">i</font>;<br/><b>take </b><font color="Maroon" title="c57">j</font> = <font color="Maroon" title="c55">j</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c56">i</font>,<font color="Maroon" title="c57">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c56">i</font>,<font color="Maroon" title="c57">j</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" title="E130">A394</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" title="E135">A401</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" title="E137">A403</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" title="E135">A404</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</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" title="E117">A300</font></i></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">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" title="E136">A405</font></i>: 
<font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c48">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">pl</font>
 ;<br/><div><i><font color="Green" title="E137">A406</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" title="c52">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c52">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font> &amp; <font color="Maroon" title="c52">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</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" title="c52">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; <font color="Maroon" title="c52">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">n</font></span>)</span> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c52">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E132">A398</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" title="E133">A399</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" title="E138">A407</font></i>: 
( <font color="Maroon" title="c49">w1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c52">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E130">A394</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c53">i1</font>, <font color="Maroon" title="c54">i2</font>, <font color="Maroon" title="c55">j1</font>, <font color="Maroon" title="c56">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c53">i1</font>,<font color="Maroon" title="c54">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c55">j1</font>,<font color="Maroon" title="c56">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c53">i1</font>,<font color="Maroon" title="c54">i2</font> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c55">j1</font>,<font color="Maroon" title="c56">j2</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" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c53">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c55">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c54">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c56">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</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" title="E138">A407</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E138">A408</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" title="c52">i1</font>, <font color="Maroon" title="c53">i2</font>, <font color="Maroon" title="c54">j1</font>, <font color="Maroon" title="c55">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E139">A409</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c52">i1</font>,<font color="Maroon" title="c53">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">j1</font>,<font color="Maroon" title="c55">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c52">i1</font>,<font color="Maroon" title="c53">i2</font> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">j1</font>,<font color="Maroon" title="c55">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</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" title="E140">A410</font></i>: 
( <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">l1</font>,<font color="Maroon" title="c41">K1</font> &amp; <font color="Maroon" title="c18">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</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" title="E111">A293</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" title="E115">A297</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">l1</font>,<font color="Maroon" title="c41">K1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E141">A411</font></i>: 
( <font color="Maroon" title="c52">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">l1</font> &amp; <font color="Maroon" title="c53">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">K1</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E139">A409</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" title="E140">A410</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">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" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E132">A398</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" title="E133">A399</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" title="E139">A409</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E142">A412</font></i>: <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E130">A394</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">mi</font>,<font color="Maroon" title="c41">K1</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</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" title="E102">A259</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">mi</font>,<font color="Maroon" title="c41">K1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="c54">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</font> &amp; <font color="Maroon" title="c55">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">K1</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" title="E139">A409</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" title="E142">A412</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c52">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c54">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c53">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c55">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">mi</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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" title="E141">A411</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><b>set </b><font color="Maroon" title="c52">hf</font> = <font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font>;<br/><div><i><font color="Green" title="E139">A413</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" title="c53">i1</font>, <font color="Maroon" title="c54">i2</font>, <font color="Maroon" title="c55">j1</font>, <font color="Maroon" title="c56">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E140">A414</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c53">i1</font>,<font color="Maroon" title="c54">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c55">j1</font>,<font color="Maroon" title="c56">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c53">i1</font>,<font color="Maroon" title="c54">i2</font> &amp; <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c55">j1</font>,<font color="Maroon" title="c56">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</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" title="E141">A415</font></i>: 
( <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c43">K2</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</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" title="E113">A295</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" title="E116">A299</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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c43">K2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E142">A416</font></i>: 
( <font color="Maroon" title="c55">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c56">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">K2</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" title="E140">A414</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" title="E141">A415</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">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" title="E143">A417</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E133">A399</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" title="E134">A400</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><i><font color="Green" title="E144">A418</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</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" title="E143">A417</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c48">LL</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" title="E132">A398</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" title="E133">A399</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" title="E143">A417</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">ma</font>,<font color="Maroon" title="c43">K2</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" title="E109">A278</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" title="E130">A394</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" title="E132">A398</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" title="E133">A399</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" title="E143">A417</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">ma</font>,<font color="Maroon" title="c43">K2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="c53">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> &amp; <font color="Maroon" title="c54">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">K2</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" title="E140">A414</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" title="E144">A418</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c53">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c55">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c54">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c56">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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" title="E142">A416</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c13">ma</font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">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" title="E140">A419</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</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" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</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" title="E118">A302</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" title="E135">A404</font></i></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">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" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> holds <br/>for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b4">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b5">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E119">A304</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" title="E137">A406</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" title="E138">A408</font></i></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">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" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E120">A307</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" title="E139">A413</font></i></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green" title="E141">A420</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" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="E142">A421</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c45">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E111">A293</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">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" title="E143">A422</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">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" title="E144">A423</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">pf</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green" title="E20">A19</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" title="E91">A248</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" title="E134">A400</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E145">A424</font></i>: 
<font color="Maroon" title="c35">pf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c36">pl</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" title="E94">A251</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" title="E129">A393</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="E132">A398</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" title="E133">A399</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" title="E143">A422</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" title="E144">A423</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E132">A398</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" title="E144">A423</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">pf</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" title="E92">A249</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" title="E145">A424</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">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" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</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" title="E91">A248</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" title="E142">A421</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" title="E142">A425</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">mi</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" title="E142">A425</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E143">A426</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">l1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="E144">A427</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c18">l1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E145">A428</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E146">A429</font></i>: 
<font color="Maroon" title="c41">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">L1</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E147">A430</font></i>: 
<font color="Maroon" title="c10">L1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c41">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">K1</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</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" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</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" title="E111">A293</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" title="E143">A426</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" title="E145">A428</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="E111">A293</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" title="E112">A294</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" title="E144">A427</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c41">K1</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" title="E111">A293</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" title="E112">A294</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" title="E144">A427</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">L1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c41">K1</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</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" title="E147">A430</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</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" title="E146">A429</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green" title="E142">A431</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">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" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</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" title="E143">A432</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</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" title="E134">A400</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><div><i><font color="Green" title="E144">A433</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" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> or <font color="Maroon" title="c13">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E145">A434</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="c46">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E113">A295</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">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" title="E146">A435</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green" title="E20">A19</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" title="E92">A249</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">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" title="E147">A436</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font> &amp; <font color="Maroon" title="c36">pl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c36">pl</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" title="E134">A400</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</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" title="E146">A435</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">LL</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" title="E132">A398</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" title="E146">A435</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" title="E147">A436</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">pl</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" title="E132">A398</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" title="E133">A399</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" title="E136">A405</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" title="E147">A436</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">pl</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" title="E92">A249</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" title="E147">A436</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">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" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E92">A249</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" title="E145">A434</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" title="E145">A437</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c13">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E145">A437</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E146">A438</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="E147">A439</font></i>: 
<font color="Maroon" title="c19">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E148">A440</font></i>: 
<font color="Maroon" title="c43">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Ll</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E149">A441</font></i>: 
<font color="Maroon" title="c11">Ll</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c43">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">K2</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="E150">A442</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</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" title="E113">A295</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" title="E146">A438</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">l2</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" title="E113">A295</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" title="E150">A442</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">l2</font></span>)</span>,<font color="Maroon" title="c43">K2</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" title="E113">A295</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" title="E114">A296</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" title="E147">A439</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">Ll</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c43">K2</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</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" title="E149">A441</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E148">A440</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green" title="E145">A443</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">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" title="E146">A444</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="c53">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E147">A445</font></i>: 
<font color="Maroon" title="c53">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E148">A446</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c54">gg</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</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" title="E149">A447</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c55">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</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" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</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" title="E145">A443</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/>
<b>then </b><i><font color="Green" title="E150">A448</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</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" title="E148">A446</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">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" title="E151">A449</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c41">K1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c43">K2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="c55">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</font></span>)</span></span>)</span> or <font color="Maroon" title="c55">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</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" title="E149">A447</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" title="E150">A448</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c55">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</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" title="E152">A450</font></i>: 
( <font color="Maroon" title="c55">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font> &amp; <font color="Maroon" title="c55">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c56">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E153">A451</font></i>: 
( <font color="Maroon" title="c56">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</font> &amp; <font color="Maroon" title="c55">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c56">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E154">A452</font></i>: 
<font color="Maroon" title="c55">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c56">i</font>,<font color="Maroon" title="c41">K1</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" title="E111">A293</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" title="E153">A451</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c57">y</font> = <font color="Maroon" title="c55">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E153">A451</font></i></a></i>;<br/><i><font color="Green" title="E155">A453</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">CK1</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E151">A449</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E156">A454</font></i>: 
<font color="Maroon" title="c56">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">CK1</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" title="E115">A297</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" title="E153">A451</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" title="E157">A455</font></i>: 
( <font color="Maroon" title="c15">Lmi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c41">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Lmi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">K1</font> &amp; <font color="Maroon" title="c42">CK1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">CK1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">mi</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</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" title="E99">A256</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" title="E155">A453</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c42">CK1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c56">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">CK1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c56">i</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" title="E156">A454</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c57">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">CK1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c56">i</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" title="E154">A452</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" title="E155">A453</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" title="E156">A454</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="c57">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c42">CK1</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" title="E156">A454</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E158">A456</font></i>: 
( <font color="Maroon" title="c41">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">k</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</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" title="E134">A400</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" title="E147">A445</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" title="E152">A450</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">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" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E130">A394</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" title="E132">A398</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" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">CK1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</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" title="E157">A455</font></i></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">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" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</font></span>)</span> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</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" title="E132">A398</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" title="E133">A399</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" title="E155">A453</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" title="E158">A456</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E148">A446</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c55">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</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" title="E152">A457</font></i>: 
( <font color="Maroon" title="c55">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font> &amp; <font color="Maroon" title="c55">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c56">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E153">A458</font></i>: 
( <font color="Maroon" title="c56">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</font> &amp; <font color="Maroon" title="c55">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c56">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E154">A459</font></i>: 
<font color="Maroon" title="c55">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c56">i</font></span>)</span>,<font color="Maroon" title="c43">K2</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" title="E113">A295</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" title="E153">A458</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c57">y</font> = <font color="Maroon" title="c55">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E153">A458</font></i></a></i>;<br/><i><font color="Green" title="E155">A460</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c44">CK2</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E151">A449</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E156">A461</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c56">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c44">CK2</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" title="E116">A299</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" title="E153">A458</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" title="E157">A462</font></i>: 
( <font color="Maroon" title="c26">Lma</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c43">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">K2</font> &amp; <font color="Maroon" title="c44">CK2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">CK2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</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" title="E105">A274</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" title="E155">A460</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c44">CK2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c56">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">CK2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c56">i</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" title="E156">A461</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c57">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">CK2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c56">i</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" title="E154">A459</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" title="E155">A460</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" title="E156">A461</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="c57">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c44">CK2</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" title="E156">A461</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E158">A463</font></i>: 
<font color="Maroon" title="c43">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">k</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</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" title="E147">A445</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" title="E152">A457</font></i></a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">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" title="E159">A464</font></i>: 
<font color="Maroon" title="c48">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</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" title="E134">A400</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c48">LL</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c48">LL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c48">LL</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" title="E130">A394</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" title="E132">A398</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" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">CK2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">ma</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</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" title="E157">A462</font></i></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">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" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">k</font></span>)</span> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</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" title="E132">A398</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" title="E133">A399</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" title="E155">A460</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" title="E158">A463</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" title="E159">A464</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E148">A446</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="E147">A465</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E128">A381</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" title="E145">A443</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</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" title="E124">A340</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</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" title="E87">A242</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" title="E88">A243</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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" title="E148">A466</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E56:1_1_5_1_2_1_1_2"><i><font color="Green" title="E77">A190</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</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" title="c53">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c54">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E149">A467</font></i>: 
( <font color="Maroon" title="c54">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font> &amp; <font color="Maroon" title="c53">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c54">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="c54">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; <font color="Maroon" title="c54">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c54">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c54">n</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" title="E132">A398</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" title="E133">A399</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" title="E149">A467</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" title="c49">w1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c54">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c54">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c54">n</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" title="E130">A394</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" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E149">A467</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green" title="E32">A34</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" title="E149">A468</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</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" title="E147">A465</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" title="E148">A466</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">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" title="E150">A469</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</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" title="E142">A431</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" title="E143">A432</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E151">A470</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c53">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</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" title="E152">A471</font></i>: 
( <font color="Maroon" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">F</font> &amp; <font color="Maroon" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</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" title="E151">A470</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c54">i1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E153">A472</font></i>: 
( <font color="Maroon" title="c54">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font> &amp; <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c54">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">x</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E154">A473</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E155">A474</font></i>: 
( <font color="Maroon" title="c50">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c54">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c54">i1</font></span>)</span> &amp; <font color="Maroon" title="c49">w1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c54">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c54">i1</font></span>)</span> &amp; <font color="Maroon" title="c49">w1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c54">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E130">A394</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" title="E132">A398</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" title="E153">A472</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c55">w</font> = <font color="Maroon" title="c49">w1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c55">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c54">i1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</font> &amp; <font color="Maroon" title="c55">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c54">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g2</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green" title="E31">A33</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" title="E152">A471</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" title="E153">A472</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" title="E155">A474</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">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" title="E156">A475</font></i>: 
( <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c55">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c54">i1</font> &amp; <font color="Maroon" title="c54">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">LL</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</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" title="E132">A398</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" title="E153">A472</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" title="E154">A473</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="c55">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c54">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c55">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c48">LL</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="E31">A33</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" title="E110">A292</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" title="E156">A475</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green" title="E16">A15</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" title="E146">A444</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" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">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" title="E151">A476</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E16">A15</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="goboard4.html#E58:1_1_5_1_2_1_1_2"><i><font color="Green" title="E79">A193</font></i></a>, <a class="txt" href="goboard4.html#E59:1_1_5_1_2_1_1_2"><i><font color="Green" title="E80">A194</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" title="E140">A419</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" title="E141">A420</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" title="E144">A433</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" title="E150">A469</font></i></a>, <a class="ref" href="goboard1.html#T37" title="GOBOARD1:th.37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41" title="GOBOARD1:th.41">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" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</font></i></a>, <a class="ref" href="goboard1.html#T26" title="GOBOARD1:th.26">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" title="E152">A477</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</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" title="E88">A243</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" title="E142">A431</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" title="E143">A432</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" title="E151">A476</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c53">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</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" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c50">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</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" title="E152">A477</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E149">A468</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" title="E129">A478</font></i>: 
<font color="Maroon" title="c36">pl</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">pf</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c47">F1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</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" title="c36">pl</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">pf</font> &amp; <font color="Maroon" title="c35">pf</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E129">A478</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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" title="c36">pl</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c48">LL</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c36">pl</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">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" title="E130">A479</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c49">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="c49">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</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" title="E131">A480</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">n</font> &amp; <font color="Maroon" title="c49">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">LL</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</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" title="c48">LL</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">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" title="c49">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E131">A480</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c49">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">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" title="E131">A481</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">n</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="c49">n</font>, <font color="Maroon" title="c50">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E132">A482</font></i>: 
<font color="Maroon" title="c49">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</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" title="E133">A483</font></i>: 
<font color="Maroon" title="c50">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c49">n</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" title="E134">A484</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">n</font> &amp; <font color="Maroon" title="c49">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">LL</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</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" title="E132">A482</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</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" title="c48">LL</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">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" title="c49">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E134">A484</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c51">w</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c49">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">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" title="E135">A485</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c49">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c48">LL</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c49">n</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" title="E134">A484</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">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" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c49">n</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" title="E94">A251</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="c51">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c35">pf</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" title="E135">A485</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">k</font> &amp; <font color="Maroon" title="c50">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E91">A248</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" title="E133">A483</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>11</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>] <b>means </b>for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</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" title="E132">A486</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>11</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">p</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" title="c49">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E133">A487</font></i>: 
<font color="Maroon" title="c49">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c50">k</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c49">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E130">A479</font></i></a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">k</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" title="c49">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c50">k</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" title="E131">A481</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" title="E133">A487</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c49">F</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E133">A488</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">LL</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> holds <br/><font color="Maroon">S<sub>11</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font>] ) )
 <b>from </b><i><a class="ref" href="goboard1.html#S1" title="GOBOARD1:sch.1">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" title="E132">A486</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" title="E134">A489</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c50">FF</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</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" title="c36">pl</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">pf</font> &amp; <font color="Maroon" title="c35">pf</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="E129">A478</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="c36">pl</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="E135">A490</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">LL</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">LL</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" title="E133">A488</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="c51">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E136">A491</font></i>: 
<font color="Maroon" title="c51">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c52">w</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c51">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E130">A479</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" title="E133">A488</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" title="E134">A489</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" title="E137">A492</font></i>: 
( <font color="Maroon" title="c51">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c51">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">w</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" title="E133">A488</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" title="E134">A489</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" title="E136">A491</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" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c51">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c51">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c51">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">w</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" title="E131">A481</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c53">i</font>, <font color="Maroon" title="c54">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E138">A493</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c53">i</font>,<font color="Maroon" title="c54">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c53">i</font>,<font color="Maroon" title="c54">j</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon" title="c55">i</font> = <font color="Maroon" title="c53">i</font>;<br/><b>take </b><font color="Maroon" title="c56">j</font> = <font color="Maroon" title="c54">j</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c55">i</font>,<font color="Maroon" title="c56">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c51">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c55">i</font>,<font color="Maroon" title="c56">j</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" title="E131">A481</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" title="E137">A492</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" title="E138">A493</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" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</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" title="E117">A300</font></i></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">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" title="E136">A494</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</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" title="E118">A302</font></i></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green" title="E137">A495</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" title="c51">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E138">A496</font></i>: 
( <font color="Maroon" title="c51">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font> &amp; <font color="Maroon" title="c51">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c52">w3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c51">n</font>, <font color="Maroon" title="c53">w2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E130">A479</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" title="E133">A488</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" title="E134">A489</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" title="c51">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; <font color="Maroon" title="c51">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c51">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">w3</font> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c53">w2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c51">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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" title="E133">A488</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" title="E134">A489</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" title="E138">A496</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" title="E139">A497</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c51">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c51">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c52">w3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c53">w2</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" title="E131">A481</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" title="E140">A498</font></i>: 
<font color="Maroon" title="c53">w2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c51">n</font>
 ;<br/><b>let </b><font color="Maroon" title="c54">i1</font>, <font color="Maroon" title="c55">i2</font>, <font color="Maroon" title="c56">j1</font>, <font color="Maroon" title="c57">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">i1</font>,<font color="Maroon" title="c55">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c56">j1</font>,<font color="Maroon" title="c57">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c51">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">i1</font>,<font color="Maroon" title="c55">i2</font> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c56">j1</font>,<font color="Maroon" title="c57">j2</font> )
 ;<br/><b>hence </b>1 = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c56">j1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c54">i1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c57">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c55">i2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green" title="E11">A10</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" title="E139">A497</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" title="E140">A498</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c54">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c56">j1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c55">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c57">j2</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c54">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c56">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c55">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c57">j2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c54">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c56">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c55">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c57">j2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">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" title="E138">A499</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">pf</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" title="c51">i1</font>, <font color="Maroon" title="c52">i2</font>, <font color="Maroon" title="c53">j1</font>, <font color="Maroon" title="c54">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E139">A500</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c51">i1</font>,<font color="Maroon" title="c52">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c53">j1</font>,<font color="Maroon" title="c54">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c51">i1</font>,<font color="Maroon" title="c52">i2</font> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c53">j1</font>,<font color="Maroon" title="c54">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</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" title="E140">A501</font></i>: 
( <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">l1</font>,<font color="Maroon" title="c41">K1</font> &amp; <font color="Maroon" title="c18">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</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" title="E111">A293</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" title="E115">A297</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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">l1</font>,<font color="Maroon" title="c41">K1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E141">A502</font></i>: 
( <font color="Maroon" title="c51">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">l1</font> &amp; <font color="Maroon" title="c52">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">K1</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E139">A500</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" title="E140">A501</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c55">w4</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><i><font color="Green" title="E142">A503</font></i>: <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c55">w4</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" title="E133">A488</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" title="E134">A489</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" title="E139">A500</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c55">w4</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" title="E131">A481</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" title="E133">A488</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" title="E134">A489</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" title="E139">A500</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">mi</font>,<font color="Maroon" title="c41">K1</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</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" title="E102">A259</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">mi</font>,<font color="Maroon" title="c41">K1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="c53">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">mi</font> &amp; <font color="Maroon" title="c54">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">K1</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" title="E139">A500</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" title="E142">A503</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c51">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c53">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c52">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c54">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c12">mi</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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" title="E141">A502</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">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" title="E139">A504</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> holds <br/>for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b4">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b5">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E119">A304</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" title="E137">A495</font></i></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></i>;<br/><b>set </b><font color="Maroon" title="c51">hf</font> = <font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</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" title="c52">i1</font>, <font color="Maroon" title="c53">i2</font>, <font color="Maroon" title="c54">j1</font>, <font color="Maroon" title="c55">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E140">A505</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c52">i1</font>,<font color="Maroon" title="c53">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">j1</font>,<font color="Maroon" title="c55">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c52">i1</font>,<font color="Maroon" title="c53">i2</font> &amp; <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">j1</font>,<font color="Maroon" title="c55">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</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" title="E141">A506</font></i>: 
( <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c43">K2</font> &amp; <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</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" title="E113">A295</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" title="E116">A299</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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c43">K2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="E142">A507</font></i>: 
( <font color="Maroon" title="c54">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c55">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">K2</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" title="E140">A505</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" title="E141">A506</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">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" title="E143">A508</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E135">A490</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c56">u</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c48">LL</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><i><font color="Green" title="E144">A509</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</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" title="E143">A508</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c56">u</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" title="E133">A488</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" title="E134">A489</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" title="E143">A508</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c56">u</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" title="E131">A481</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" title="E133">A488</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" title="E134">A489</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" title="E143">A508</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">ma</font>,<font color="Maroon" title="c43">K2</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</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" title="E108">A277</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">ma</font>,<font color="Maroon" title="c43">K2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green" title="E56">A85</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">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" title="c52">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ma</font> &amp; <font color="Maroon" title="c53">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">K2</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" title="E140">A505</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" title="E144">A509</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c52">i1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c54">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c53">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c55">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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" title="E142">A507</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c13">ma</font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">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" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E120">A307</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" title="E139">A504</font></i></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">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" title="E140">A510</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</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" title="E136">A494</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><div><i><font color="Green" title="E141">A511</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" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="E142">A512</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c45">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E111">A293</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">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" title="E143">A513</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">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" title="E144">A514</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">pl</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green" title="E20">A19</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" title="E92">A249</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" title="E135">A490</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E145">A515</font></i>: 
<font color="Maroon" title="c35">pf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c35">pf</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" title="E94">A251</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="E133">A488</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" title="E134">A489</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" title="E143">A513</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" title="E144">A514</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">pf</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" title="E133">A488</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" title="E138">A499</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" title="E144">A514</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">pf</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" title="E91">A248</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" title="E145">A515</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">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" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</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" title="E91">A248</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" title="E142">A512</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" title="E142">A516</font></i>: 
<font color="Maroon" title="c12">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">mi</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" title="E142">A516</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E143">A517</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">l1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="E144">A518</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c18">l1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E145">A519</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E146">A520</font></i>: 
<font color="Maroon" title="c41">K1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">L1</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E147">A521</font></i>: 
<font color="Maroon" title="c10">L1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c41">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">K1</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</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" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</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" title="E111">A293</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" title="E143">A517</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" title="E145">A519</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="E111">A293</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" title="E112">A294</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" title="E144">A518</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c41">K1</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" title="E111">A293</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" title="E112">A294</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" title="E144">A518</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">L1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c41">K1</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</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" title="E147">A521</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</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" title="E146">A520</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green" title="E142">A522</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">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" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</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" title="E143">A523</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</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" title="E135">A490</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><div><i><font color="Green" title="E144">A524</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" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> or <font color="Maroon" title="c13">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E145">A525</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="c46">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E113">A295</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">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" title="E146">A526</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">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" title="E147">A527</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c36">pl</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">pl</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" title="E148">A528</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font> &amp; <font color="Maroon" title="c36">pl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c35">pf</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" title="E94">A251</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" title="E129">A478</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" title="E135">A490</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">F</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" title="E146">A526</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">LL</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" title="E133">A488</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" title="E146">A526</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" title="E148">A528</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">pl</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" title="E133">A488</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" title="E134">A489</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" title="E147">A527</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" title="E148">A528</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c36">pl</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" title="E91">A248</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" title="E148">A528</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">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" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E92">A249</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" title="E145">A525</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" title="E145">A529</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="c13">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c13">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E145">A529</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E146">A530</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">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" title="E147">A531</font></i>: 
<font color="Maroon" title="c19">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="E148">A532</font></i>: 
<font color="Maroon" title="c43">K2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Ll</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E149">A533</font></i>: 
<font color="Maroon" title="c11">Ll</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c43">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">K2</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="E150">A534</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</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" title="E113">A295</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" title="E146">A530</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">l2</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" title="E113">A295</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" title="E150">A534</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">l2</font></span>)</span>,<font color="Maroon" title="c43">K2</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" title="E113">A295</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" title="E114">A296</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" title="E147">A531</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">Ll</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c43">K2</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</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" title="E149">A533</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E148">A532</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green" title="E145">A535</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">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" title="E146">A536</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="c52">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E147">A537</font></i>: 
<font color="Maroon" title="c52">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E148">A538</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c53">gg</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</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" title="E149">A539</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c54">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</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" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</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" title="E145">A535</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/>
<b>then </b><i><font color="Green" title="E150">A540</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</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" title="E148">A538</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">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" title="E151">A541</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c41">K1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c43">K2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="c54">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</font></span>)</span></span>)</span> or <font color="Maroon" title="c54">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</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" title="E149">A539</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" title="E150">A540</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c54">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</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" title="E152">A542</font></i>: 
( <font color="Maroon" title="c54">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font> &amp; <font color="Maroon" title="c54">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c55">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E153">A543</font></i>: 
( <font color="Maroon" title="c55">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">h1</font> &amp; <font color="Maroon" title="c54">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c55">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E154">A544</font></i>: 
<font color="Maroon" title="c54">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c55">i</font>,<font color="Maroon" title="c41">K1</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" title="E111">A293</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" title="E153">A543</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c56">y</font> = <font color="Maroon" title="c54">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E153">A543</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" title="E155">A545</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">pf</font>
 ;<br/><i><font color="Green" title="E156">A546</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">CK1</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E151">A541</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E157">A547</font></i>: 
<font color="Maroon" title="c55">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">CK1</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" title="E115">A297</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" title="E153">A543</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" title="E158">A548</font></i>: 
( <font color="Maroon" title="c15">Lmi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c41">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Lmi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">K1</font> &amp; <font color="Maroon" title="c42">CK1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">CK1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">mi</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</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" title="E99">A256</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" title="E156">A546</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c42">CK1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">CK1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">i</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" title="E157">A547</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c56">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">CK1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c55">i</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" title="E154">A544</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" title="E156">A546</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" title="E157">A547</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="c56">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c42">CK1</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" title="E157">A547</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E159">A549</font></i>: 
( <font color="Maroon" title="c41">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">k</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</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" title="E135">A490</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" title="E147">A537</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" title="E152">A542</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">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" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">pf</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c35">pf</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" title="E131">A481</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" title="E133">A488</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" title="E155">A545</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" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">CK1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">mi</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</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" title="E99">A256</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" title="E100">A257</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" title="E101">A258</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" title="E158">A548</font></i></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">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" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</font></span>)</span> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green" title="E23">A25</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" title="E133">A488</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" title="E134">A489</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" title="E156">A546</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" title="E159">A549</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E148">A538</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c54">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</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" title="E152">A550</font></i>: 
( <font color="Maroon" title="c54">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c46">h2</font> &amp; <font color="Maroon" title="c54">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c55">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E153">A551</font></i>: 
( <font color="Maroon" title="c55">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c46">h2</font> &amp; <font color="Maroon" title="c54">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c55">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E154">A552</font></i>: 
<font color="Maroon" title="c54">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c55">i</font></span>)</span>,<font color="Maroon" title="c43">K2</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" title="E113">A295</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" title="E153">A551</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c56">y</font> = <font color="Maroon" title="c54">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">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" title="E153">A551</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" title="E155">A553</font></i>: 
<font color="Maroon" title="c48">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c48">LL</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" title="E135">A490</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c57">u</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c48">LL</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><i><font color="Green" title="E156">A554</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c44">CK2</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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" title="E151">A541</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="E157">A555</font></i>: 
<font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c44">CK2</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" title="E116">A299</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" title="E153">A551</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" title="E158">A556</font></i>: 
( <font color="Maroon" title="c26">Lma</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c43">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">K2</font> &amp; <font color="Maroon" title="c44">CK2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">CK2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ma</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</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" title="E105">A274</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" title="E156">A554</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c44">CK2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c55">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">CK2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c55">i</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" title="E157">A555</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="c56">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">CK2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c55">i</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" title="E154">A552</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" title="E156">A554</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" title="E157">A555</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="c56">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c44">CK2</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" title="E157">A555</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E159">A557</font></i>: 
<font color="Maroon" title="c43">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">k</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</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" title="E147">A537</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" title="E152">A550</font></i></a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">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" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c57">u</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c57">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c57">u</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" title="E131">A481</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" title="E133">A488</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" title="E155">A553</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" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">CK2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">ma</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</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" title="E105">A274</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" title="E106">A275</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" title="E107">A276</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" title="E158">A556</font></i></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">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" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c52">k</font></span>)</span> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c48">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green" title="E25">A27</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" title="E133">A488</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" title="E134">A489</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" title="E155">A553</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" title="E156">A554</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" title="E159">A557</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="E148">A538</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="E147">A558</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c45">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E128">A381</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" title="E145">A535</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</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" title="E124">A340</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</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" title="E87">A242</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" title="E88">A243</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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" title="E148">A559</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E56:1_1_5_1_2_1_1_2"><i><font color="Green" title="E77">A190</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</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" title="c52">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c52">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c53">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E149">A560</font></i>: 
( <font color="Maroon" title="c53">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font> &amp; <font color="Maroon" title="c52">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c53">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c54">u</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c53">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E130">A479</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" title="E133">A488</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" title="E134">A489</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" title="E149">A560</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" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c54">u</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" title="E133">A488</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" title="E134">A489</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" title="E149">A560</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" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c54">u</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" title="E131">A481</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" title="E133">A488</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" title="E134">A489</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" title="E149">A560</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" title="c52">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E149">A560</font></i></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">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" title="c52">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green" title="E32">A34</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" title="E149">A561</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</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" title="E147">A558</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" title="E148">A559</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">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" title="E150">A562</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</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" title="E142">A522</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" title="E143">A523</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E151">A563</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c52">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</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" title="E152">A564</font></i>: 
( <font color="Maroon" title="c52">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c49">F</font> &amp; <font color="Maroon" title="c52">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</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" title="E151">A563</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c53">i1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E153">A565</font></i>: 
( <font color="Maroon" title="c53">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">F</font> &amp; <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c53">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">x</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c54">w</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c53">i1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="E130">A479</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" title="E133">A488</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" title="E134">A489</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" title="E153">A565</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" title="E154">A566</font></i>: 
( <font color="Maroon" title="c49">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c53">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c54">w</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c35">pf</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c54">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c54">w</font> &amp; <font color="Maroon" title="c54">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</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" title="E131">A481</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" title="E133">A488</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" title="E134">A489</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" title="E153">A565</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">i1</font> &amp; <font color="Maroon" title="c53">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c48">LL</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" title="E133">A488</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" title="E153">A565</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="E155">A567</font></i>: 
( <font color="Maroon" title="c54">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c35">pf</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c48">LL</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c54">w</font> )
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">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" title="E156">A568</font></i>: 
<font color="Maroon" title="c54">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g2</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green" title="E31">A33</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" title="E154">A566</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">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" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c54">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">C</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green" title="E31">A33</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" title="E152">A564</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" title="E153">A565</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" title="E154">A566</font></i></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">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" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c54">w</font> &amp; <font color="Maroon" title="c54">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green" title="E31">A33</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" title="E104">A273</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" title="E155">A567</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" title="E156">A568</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green" title="E16">A15</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" title="E146">A536</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" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">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" title="E151">A569</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="E16">A15</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="goboard4.html#E58:1_1_5_1_2_1_1_2"><i><font color="Green" title="E79">A193</font></i></a>, <a class="txt" href="goboard4.html#E59:1_1_5_1_2_1_1_2"><i><font color="Green" title="E80">A194</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" title="E140">A510</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" title="E141">A511</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" title="E144">A524</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" title="E150">A562</font></i></a>, <a class="ref" href="goboard1.html#T37" title="GOBOARD1:th.37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41" title="GOBOARD1:th.41">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" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" title="GOBOARD1:func.4">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</font></i></a>, <a class="ref" href="goboard1.html#T26" title="GOBOARD1:th.26">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" title="E152">A570</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green" title="E39">A66</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" title="E88">A243</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" title="E142">A522</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" title="E143">A523</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" title="E151">A569</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c52">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</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" title="c52">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c49">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c46">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c34">tt</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" title="E152">A570</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="E149">A561</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:1"/><i><font color="Green" title="E5">A571</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="goboard4.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="goboard4.html#E3:1"><i><font color="Green" title="E4">A3</font></i></a>);<br/></i>
<div><i><font color="Green" title="E6">A572</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" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">G1</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/><b>let </b><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:1_2"/>
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1"><i><font color="Green" title="E5">A571</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:1"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E7:1"/><b>then </b>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/>
<b>hence </b><a NAME="E8:1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="goboard4.html#E5:1"><i><font color="Green" title="E6">A572</font></i></a></i>;<br/>


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