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

<b>let </b><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">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>let </b><font color="Maroon" title="c3">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/><b>let </b><font color="Maroon" title="c4">f</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><b>that </b><br/><a NAME="E1:57"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c1">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="c3">G</font>
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c3">G</font>
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c4">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="c4">f</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="c3">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="c3">G</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:57"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c2">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="c4">f</font>
 <b>and </b><br/><a NAME="E5:57"/><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c2">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="c3">G</font>,<font color="Maroon" title="c1">k</font></span>)</span>
 ;<br/>

<a NAME="E6:57"/><i><font color="Green" title="E45">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">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="c3">G</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard1.html#E1:57"><i><font color="Green" title="E40">A1</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="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:57"/><i><font color="Green" title="E46">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">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="c4">f</font> )
 
<b>by </b><i><a class="txt" href="goboard1.html#E4:57"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<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>( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <font color="Maroon" title="c3">G</font> implies  ex <font color="Olive" title="b1">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/>( <font color="Olive" title="b1">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="c4">f</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</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="c3">G</font>,$1</span>)</span> ) );<br/>
<a NAME="E8:57"/><i><font color="Green" title="E47">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="goboard1.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:57"/><i><font color="Green" title="E48">A9</font></i>: 
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>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</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="E1:57_1"/><i><font color="Green" title="E49">A10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font>]
 ;<br/>

<b>assume </b><a NAME="E2:57_1"/><i><font color="Green" title="E50">A11</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">i</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="c3">G</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_1_1"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="goboard1.html#E2:57_1"><i><font color="Green" title="E50">A11</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="57_1_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_1_1_1"/><i><font color="Green" title="E51">A12</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c6">j</font> = <font color="Maroon" title="c2">n</font>;<br/><b>thus </b><a NAME="E2:57_1_1_1_1"/>
( <font color="Maroon" title="c6">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="c4">f</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">j</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="c3">G</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">i</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="goboard1.html#E4:57"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="goboard1.html#E5:57"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="goboard1.html#E1:57_1_1_1_1"><i><font color="Green" title="E51">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_1_1_2"/><i><font color="Green" title="E51">A13</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:57_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <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="E3:57_1_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</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="c3">G</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> )
 <b>by </b><i><a class="txt" href="goboard1.html#E6:57"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="goboard1.html#E2:57_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:57_1_1_1_2"/><b>then </b><i><font color="Green" title="E52">A14</font></i>: 
( <font color="Maroon" title="c5">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="c3">G</font> &amp; <font color="Maroon" title="c5">i</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="c3">G</font> )
 <b>by </b><i><a class="txt" href="goboard1.html#E2:57_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</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="c4">f</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; <font color="Maroon" title="c4">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="c3">G</font>,<font color="Maroon" title="c5">i</font></span>)</span> );<br/><a NAME="E5:57_1_1_1_2"/><i><font color="Green" title="E53">A15</font></i>: 
 ex <font color="Olive" title="b1">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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>]
 <b>by </b><i><a class="txt" href="goboard1.html#E1:57_1"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="goboard1.html#E2:57_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="goboard1.html#E1:57_1_1_1_2"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:57_1_1_1_2"/><i><font color="Green" title="E54">A16</font></i>: 
for <font color="Olive" title="b1">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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/><font color="Olive" title="b1">j</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="c4">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>consider </b><font color="Maroon" title="c6">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="E8:57_1_1_1_2"/><i><font color="Green" title="E55">A17</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">ma</font>] &amp; ( for <font color="Olive" title="b1">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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">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="goboard1.html#E6:57_1_1_1_2"><i><font color="Green" title="E54">A16</font></i></a>, <a class="txt" href="goboard1.html#E5:57_1_1_1_2"><i><font color="Green" title="E53">A15</font></i></a>);<br/></i><a NAME="E9:57_1_1_1_2"/><i><font color="Green" title="E56">A18</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="c4">f</font>
 <b>by </b><i><a class="txt" href="goboard1.html#E7:57"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><i><font color="Green" title="E57">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">j</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="E1:57_1_1_1_2_1"/><i><font color="Green" title="E58">A20</font></i>: 
( <font color="Maroon" title="c7">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="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</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="c3">G</font>,<font color="Maroon" title="c5">i</font></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> or <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:57_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">ma</font>
 <b>by </b><i><a class="txt" href="goboard1.html#E8:57_1_1_1_2"><i><font color="Green" title="E55">A17</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="57_1_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:57_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">ma</font>
 <b>by </b><i><a class="txt" href="goboard1.html#E8:57_1_1_1_2"><i><font color="Green" title="E55">A17</font></i></a>, <a class="txt" href="goboard1.html#E1:57_1_1_1_2_1"><i><font color="Green" title="E58">A20</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="E3:57_1_1_1_2_1"/>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">ma</font>
 ;<br/></div><b>end;</b></div><b>take </b><font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c6">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>thus </b><a NAME="E11:57_1_1_1_2"/>
<font color="Maroon" title="c7">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="c4">f</font>
 <b>by </b><i><a class="txt" href="goboard1.html#E2:57"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="goboard1.html#E3:57"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="goboard1.html#E4:57_1_1_1_2"><i><font color="Green" title="E52">A14</font></i></a>, <a class="txt" href="goboard1.html#E8:57_1_1_1_2"><i><font color="Green" title="E55">A17</font></i></a>, <a class="txt" href="goboard1.html#E9:57_1_1_1_2"><i><font color="Green" title="E56">A18</font></i></a>, <a class="txt" href="goboard1.html#E10:57_1_1_1_2"><i><font color="Green" title="E57">A19</font></i></a>, <a class="ref" href="goboard1.html#T44" target="_self" title="GOBOARD1:th.44">Th44</a></i>;<br/><a NAME="E12:57_1_1_1_2"/>
<font color="Maroon" title="c6">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E13:57_1_1_1_2"/>
( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</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="c3">G</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">i</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="goboard1.html#E2:57"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="goboard1.html#E3:57"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="goboard1.html#E4:57_1_1_1_2"><i><font color="Green" title="E52">A14</font></i></a>, <a class="txt" href="goboard1.html#E8:57_1_1_1_2"><i><font color="Green" title="E55">A17</font></i></a>, <a class="txt" href="goboard1.html#E9:57_1_1_1_2"><i><font color="Green" title="E56">A18</font></i></a>, <a class="txt" href="goboard1.html#E10:57_1_1_1_2"><i><font color="Green" title="E57">A19</font></i></a>, <a class="ref" href="goboard1.html#T44" target="_self" title="GOBOARD1:th.44">Th44</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><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:57_1"/>
 ex <font color="Olive" title="b1">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/>( <font color="Olive" title="b1">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="c4">f</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</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="c3">G</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E10:57"/><i><font color="Green" title="E49">A21</font></i>: 
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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</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="goboard1.html#E8:57"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="goboard1.html#E9:57"><i><font color="Green" title="E48">A9</font></i></a>);</i><br/>

<b>let </b><font color="Maroon" title="c5">i</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="E11:57"/><i><font color="Green" title="E50">A22</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</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="c3">G</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">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="E13:57"/><i><font color="Green" title="E51">A23</font></i>: 
( <font color="Maroon" title="c6">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="c4">f</font> &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">j</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="c3">G</font>,<font color="Maroon" title="c5">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard1.html#E10:57"><i><font color="Green" title="E49">A21</font></i></a></i>;<br/>
<a NAME="E14:57"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="txt" href="goboard1.html#E6:57"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="goboard1.html#E11:57"><i><font color="Green" title="E50">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:57"/><b>then </b><i><font color="Green" title="E52">A24</font></i>: 
<font color="Maroon" title="c5">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="c3">G</font>
 
<b>by </b><i><a class="txt" href="goboard1.html#E11:57"><i><font color="Green" title="E50">A22</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">j</font>
;<br/>

<b>thus </b><a NAME="E16:57"/>
<font color="Maroon" title="c6">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="c4">f</font>
 <b>by </b><i><a class="txt" href="goboard1.html#E13:57"><i><font color="Green" title="E51">A23</font></i></a></i>;<br/>

<a NAME="E17:57"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font>
 
<b>by </b><i><a class="txt" href="goboard1.html#E1:57"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="goboard1.html#E5:57"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="goboard1.html#E11:57"><i><font color="Green" title="E50">A22</font></i></a>, <a class="txt" href="goboard1.html#E13:57"><i><font color="Green" title="E51">A23</font></i></a>, <a class="txt" href="goboard1.html#E15:57"><i><font color="Green" title="E52">A24</font></i></a>, <a class="ref" href="goboard1.html#T19" target="_self" title="GOBOARD1:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E18:57"/>
( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">j</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="c3">G</font>,<font color="Maroon" title="c5">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard1.html#E13:57"><i><font color="Green" title="E51">A23</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>


</div>
