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

<b>let </b><font color="Maroon">f1</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="sprect_2.html#V1">clockwise_oriented</a> <a href="goboard5.html#NM1">special_circular_sequence</a>, <font color="Maroon">f2</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="sprect_2.html#V1">clockwise_oriented</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>

<b>assume </b><a NAME="E1:33_1_2"/><i><font color="Green">E969</font></i>: 
<font color="Maroon">f1</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 ;<br/>

<b>given </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:33_1_2"/><i><font color="Green">E970</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:33_1_2"/><i><font color="Green">E971</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E4:33_1_2"/><i><font color="Green">E972</font></i>: 
 <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E5:33_1_2"/><i><font color="Green">E973</font></i>: 
 <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E6:33_1_2"/><i><font color="Green">E974</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> holds <br/>( (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f1</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f1</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> implies <font color="Maroon">f1</font> <a href="gobrd13.html#R1">turns_right</a> <font color="Olive">k</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f1</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f1</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies <font color="Maroon">f1</font> <a href="gobrd13.html#R3">goes_straight</a> <font color="Olive">k</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f1</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies <font color="Maroon">f1</font> <a href="gobrd13.html#R2">turns_left</a> <font color="Olive">k</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) )
 <b>and </b><br/><a NAME="E7:33_1_2"/><i><font color="Green">E975</font></i>: 
<font color="Maroon">f2</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 ;<br/>

<b>given </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E8:33_1_2"/><i><font color="Green">E976</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E9:33_1_2"/><i><font color="Green">E977</font></i>: 
( <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E10:33_1_2"/><i><font color="Green">E978</font></i>: 
 <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E11:33_1_2"/><i><font color="Green">E979</font></i>: 
 <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<br/>

<b>assume </b><a NAME="E12:33_1_2"/><i><font color="Green">E980</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> holds <br/>( (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f2</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f2</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> implies <font color="Maroon">f2</font> <a href="gobrd13.html#R1">turns_right</a> <font color="Olive">k</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f2</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f2</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies <font color="Maroon">f2</font> <a href="gobrd13.html#R3">goes_straight</a> <font color="Olive">k</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f2</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies <font color="Maroon">f2</font> <a href="gobrd13.html#R2">turns_left</a> <font color="Olive">k</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E13:33_1_2"/><i><font color="Green">E981</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E14:33_1_2"/><i><font color="Green">E982</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_2_1"/><i><font color="Green">E983</font></i>: 
<font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:33_1_2_1"/><i><font color="Green">E984</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_5.html#T23">FINSEQ_5:23</a></i>;<br/>
<a NAME="E3:33_1_2_1"/><i><font color="Green">E985</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 
<b>by </b><i>, , <a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E4:33_1_2_1"/><i><font color="Green">E986</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4 )
 
<b>by </b><i><a class="ref" href="goboard7.html#T36">GOBOARD7:36</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_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="E1:33_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_2_1_1_1"/>
<font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_2"/><i><font color="Green">E987</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:33_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="jordan8.html#T1">JORDAN8:1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_2_1_1_2"/>
<font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3"/><i><font color="Green">E988</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_3"/><i><font color="Green">E989</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_2_1_1_3_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3_1_1_1"/><i><font color="Green">E990</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">m</font> = <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E2:33_1_2_1_1_3_1_1_1"/><i><font color="Green">E991</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E3:33_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E992</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T6">JORDAN3:6</a></i>;<br/><div><i><font color="Green">E993</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_2_1_1_3_1_1_1_1"/><i><font color="Green">E994</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:33_1_2_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green">E995</font></i>: 
<font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><a NAME="E3:33_1_2_1_1_3_1_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E4:33_1_2_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green">E996</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><a NAME="E5:33_1_2_1_1_3_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E6:33_1_2_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green">E997</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><a NAME="E7:33_1_2_1_1_3_1_1_1_1"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> )
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_5.html#T18">FINSEQ_5:18</a></i>;<br/><b>hence </b><a NAME="E8:33_1_2_1_1_3_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="goboard7.html#T40">GOBOARD7:40</a></i>;<br/></div><b>end;</b></div><a NAME="E5:33_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E998</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:33_1_2_1_1_3_1_1_1"/><i><font color="Green">E999</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:33_1_2_1_1_3_1_1_1"/><i><font color="Green">E1000</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , <a class="ref" href="gobrd13.html#T43">GOBRD13:43</a></i>;<br/><a NAME="E8:33_1_2_1_1_3_1_1_1"/><i><font color="Green">E1001</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , <a class="ref" href="gobrd13.html#T43">GOBRD13:43</a></i>;<br/><a NAME="E9:33_1_2_1_1_3_1_1_1"/><i><font color="Green">E1002</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f2</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="gobrd13.html#T43">GOBRD13:43</a></i>;<br/><a NAME="E10:33_1_2_1_1_3_1_1_1"/><i><font color="Green">E1003</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f2</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="gobrd13.html#T43">GOBRD13:43</a></i>;<br/><a NAME="E11:33_1_2_1_1_3_1_1_1"/><i><font color="Green">E1004</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_2_1_1_3_1_1_1_2_1"/>
( (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) or (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) or  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3_1_1_1_2_1_1"/>
(  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_3_1_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">f1</font> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">f2</font> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T3" target="_self">Th3</a>, , , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_2_1_1_3_1_1_1_2_1_1"/>
<font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , , <a class="ref" href="gobrd13.html#T47">GOBRD13:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3_1_1_1_2_1_2"/>
(  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_3_1_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="gobrd13.html#R3">goes_straight</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">f2</font> <a href="gobrd13.html#R3">goes_straight</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T3" target="_self">Th3</a>, , , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_2_1_1_3_1_1_1_2_1_2"/>
<font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , , <a class="ref" href="gobrd13.html#T49">GOBRD13:49</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3_1_1_1_2_1_3"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_3_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon">f1</font> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">f2</font> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T3" target="_self">Th3</a>, , , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , </i>;<br/><b>hence </b><a NAME="E3:33_1_2_1_1_3_1_1_1_2_1_3"/>
<font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , , <a class="ref" href="gobrd13.html#T48">GOBRD13:48</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:33_1_2_1_1_3_1_1_1"/>
<font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3_1_1_2"/><i><font color="Green">E1005</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E1006</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><a NAME="E3:33_1_2_1_1_3_1_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E4:33_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E1007</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><a NAME="E5:33_1_2_1_1_3_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E6:33_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E1008</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><a NAME="E7:33_1_2_1_1_3_1_1_2"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_5.html#T18">FINSEQ_5:18</a></i>;<br/><a NAME="E8:33_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E1009</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , <a class="ref" href="goboard7.html#T40">GOBOARD7:40</a></i>;<br/><a NAME="E9:33_1_2_1_1_3_1_1_2"/><i><font color="Green">E1010</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">f1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon">f2</font>
<b>by </b><i>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_2_1_1_3"/>
<font color="Maroon">f1</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E15:33_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E16:33_1_2"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i/>;<br/>


</div>
